An explicit trace formula of Jacquet-Zagier type for Hilbert modular forms

Shingo Sugiyama  and  Masao Tsuzuki
Abstract.

We give an exact formula of the average of adjoint L𝐿L-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier’s formula known for the case of elliptic cusp forms on SL2()subscriptSL2{\operatorname{SL}}_{2}({\mathbb{Z}}). As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint L𝐿L-functions.

Key words and phrases:
trace formulas, adjoint L𝐿L-functions.
2010 Mathematics Subject Classification:
Primary 11F72; Secondary 11F67.

1. Introduction

1.1. Background and motivation

In [35], Zagier proposed an elegant way to compute the traces of Hecke operators on the space of elliptic cusp forms by means of the Rankin-Selberg method. This is a more direct way than the Selberg’s one because it does not need a deliberate rearrangement for renormalization of divergent terms which are inevitably produced by truncation process. Later, a similar study was conducted for Maass forms on SL2()subscriptSL2{\operatorname{SL}}_{2}({\mathbb{Z}}) in [36], and for general cusp forms on the adelization GL2(𝔸F)subscriptGL2subscript𝔸𝐹{\operatorname{GL}}_{2}({\mathbb{A}}_{F}) with an arbitrary number field F𝐹F in the work of Jacquet and Zagier [12]. The formula proved in [12] can be viewed as an “arithmetic deformation” by a complex parameter z𝑧z of the usual Arthur-Selberg trace formula for GL(2)GL2{\operatorname{GL}}(2) because the latter one is expected to be recovered from the former one as the residue at z=1𝑧1z=1. Although their general formula is less explicit than Zagier’s one, it provides us with a different proof of holomorphicity of the symmetric square L𝐿L-function of a cuspidal representation of GL2(𝔸F)subscriptGL2subscript𝔸𝐹{\operatorname{GL}}_{2}({\mathbb{A}}_{F}), which was first proved by Shimura [22] in a classical setting and later was generalized to an adelic setting by [6]. The paper [35] also gave an application of the formula to the proof of the algebraicity of the critical values of the symmetric square L𝐿L-functions for elliptic cusp forms, which was independently obtained by Strum [23] based on [22]. Mizumoto [18] and Takase [29] extended Zagier’s method to Hilbert cusp forms under the assumption that the narrow class number of the base field is one. For application to special values, the explicit nature of their formula is crucial.

In this paper, motivated by these works and intending further potential applications, we shall calculate Jacquet-Zagier’s trace formula for holomorphic Hilbert cusp forms as explicitly as possible when the level is square-free without the assumption on the class number of the base field F𝐹F. For a technical reason, we assume that the prime 222 splits completely in F𝐹F. We remark that this assumption is mild enough to include the interesting cases F=𝐹F=\mathbb{Q} and F=(D)𝐹𝐷F=\mathbb{Q}(\sqrt{D}) with D>0𝐷0D>0 and D1(mod 8)𝐷1mod 8D\equiv 1(\text{mod }8), and our formulas for Hilbert modular forms with large levels are new even for F=𝐹F=\mathbb{Q}. Since we use the matrix coefficients of discrete series representations at archimedean places, which are not compactly supported contrary to the test functions dealt in [12], we have to modify the convergence proof of the geometric side in a substantial way. Moreover, we completely calculate all local terms in the formula for a large class of test functions. As an application, we prove an equidistribution result of Satake parameters in the ensemble defined by the symmetric square L𝐿L-functions L(z,π;Ad)𝐿𝑧𝜋AdL(z,\pi;{\operatorname{Ad}}) of holomorphic Hilbert cusp forms π𝜋\pi of a fixed weight with the varying square-free levels, in such a way that a part of the famous Serre’s equidistribution theorem ([19, Théorème 1]) is recovered from our formula by the specialization at z=1𝑧1z=1. The non-vanishing of the symmetric square L𝐿L-function of an elliptic modular form at a point in the critical strip has been pursued by many authors ([13], [15], [2]). As a corollary to our asymptotic formula, we obtain infinitely many Hilbert cusp forms with a fixed weight of growing levels whose symmetric square L𝐿L-functions are non-vanishing at a given point in the critical strip. Our method is not the Rankin-Selberg method in the accurate sence, because the Eisenstein series is not unfolded on the convergence region as was done in [35] and [12]; actually, the same proof works if the Eisenstein series is replaced with a Maass cusp form. To illustrate the robustness of our method, we deduce Theorem 9.1 which is viewed as an adelic version of Gon’s formula in an opposite setting to [8].

1.2. Description of main results

Let us state our main result introducing notation used in this article on the way. Let \mathbb{N} be the set of all positive integers and we write 0subscript0\mathbb{N}_{0} for {0}0\mathbb{N}\cup\{0\}. For any condition P𝑃P, we put δ(P)=1𝛿𝑃1\delta(P)=1 if P𝑃P is true, and δ(P)=0𝛿𝑃0\delta(P)=0 if P𝑃P is false, respectively. We set Γ(s)=πs/2Γ(s/2)subscriptΓ𝑠superscript𝜋𝑠2Γ𝑠2\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2) and Γ(s)=2(2π)sΓ(s)subscriptΓ𝑠2superscript2𝜋𝑠Γ𝑠\Gamma_{{\mathbb{C}}}(s)=2(2\pi)^{-s}\Gamma(s). All the fractional ideals appearing in this paper are supposed to be invertible.

Let F𝐹F be a totally real number field, 𝔬𝔬\mathfrak{o} the integer ring of F𝐹F and 𝔸𝔸{\mathbb{A}} the adele ring of F𝐹F. The completed Dedekind zeta function of F𝐹F is denoted by ζF(s)subscript𝜁𝐹𝑠\zeta_{F}(s). (All L𝐿L-functions in this article are supposed to be completed by appropriate gamma factors.) The set of finite places and the set of archimedean places of F𝐹F are denoted by ΣfinsubscriptΣfin\Sigma_{\rm fin} and ΣsubscriptΣ\Sigma_{\infty}, respectively. Set ΣF=ΣfinΣsubscriptΣ𝐹subscriptΣfinsubscriptΣ\Sigma_{F}=\Sigma_{\rm fin}\cup\Sigma_{\infty}. The completion of F𝐹F at vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F} is denoted by Fvsubscript𝐹𝑣F_{v} and the modulus of Fvsubscript𝐹𝑣F_{v} is by ||v|\,|_{v}. Then, ||𝔸=vΣF||v|\,|_{\mathbb{A}}=\prod_{v\in\Sigma_{F}}|\,|_{v} defines the idele norm on 𝔸×superscript𝔸\mathbb{A}^{\times}. When vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}, 𝔬vsubscript𝔬𝑣\mathfrak{o}_{v}, 𝔭vsubscript𝔭𝑣{\mathfrak{p}}_{v} and qvsubscript𝑞𝑣q_{v} denote the maximal order of Fvsubscript𝐹𝑣F_{v}, the maximal ideal of 𝔬vsubscript𝔬𝑣\mathfrak{o}_{v} and #(𝔬v/𝔭v)#subscript𝔬𝑣subscript𝔭𝑣\#({\mathfrak{o}}_{v}/{\mathfrak{p}}_{v}), respectively. For a non-zero ideal 𝔞𝔬𝔞𝔬{\mathfrak{a}}\subset{\mathfrak{o}}, let N(𝔞)=vΣfinqvordv(𝔞)N𝔞subscriptproduct𝑣subscriptΣfinsuperscriptsubscript𝑞𝑣subscriptord𝑣𝔞{\operatorname{N}}({\mathfrak{a}})=\prod_{v\in\Sigma_{\rm fin}}q_{v}^{\operatorname{ord}_{v}({\mathfrak{a}})} denote the absolute norm of 𝔞𝔞{\mathfrak{a}} and S(𝔞)𝑆𝔞S({\mathfrak{a}}) the set of vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} dividing 𝔞𝔞{\mathfrak{a}}. Let DFsubscript𝐷𝐹D_{F} be the absolute discriminant of F/𝐹F/{\mathbb{Q}} and ψ=vΣFψv\psi=\otimes_{v\in\Sigma_{F}}\psi_{v} the character of 𝔸/F𝔸𝐹{\mathbb{A}}/F defined as ψtrF/subscript𝜓subscripttr𝐹\psi_{{\mathbb{Q}}}\circ{\operatorname{tr}}_{F/{\mathbb{Q}}} where ψsubscript𝜓\psi_{\mathbb{Q}} is the character of 𝔸/subscript𝔸{\mathbb{A}}_{\mathbb{Q}}/\mathbb{Q} such that ψ(x)=e2πixsubscript𝜓𝑥superscript𝑒2𝜋𝑖𝑥\psi_{\mathbb{Q}}(x)=e^{2\pi ix} for x𝑥x\in{\mathbb{R}}. Let B𝐵B, H𝐻H and N𝑁N be F𝐹F-subgroups of G=GL(2)𝐺GL2G={\operatorname{GL}}(2) defined symbolically as B={[0]}𝐵delimited-[]0B=\{\left[\begin{smallmatrix}*&*\\ 0&*\end{smallmatrix}\right]\}, H={[00]}𝐻delimited-[]00H=\{\left[\begin{smallmatrix}*&0\\ 0&*\end{smallmatrix}\right]\} and N={[101]}𝑁delimited-[]101N=\{\left[\begin{smallmatrix}1&*\\ 0&1\end{smallmatrix}\right]\}; B=HN𝐵𝐻𝑁B=HN is a Borel subgroup of G𝐺G. Let Z𝑍Z be the center of G𝐺G. For vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}, let 𝕂vsubscript𝕂𝑣{\mathbb{K}}_{v} denote G(𝔬v)𝐺subscript𝔬𝑣G(\mathfrak{o}_{v}) if vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and O(2)O2{\rm O}(2) if vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Set 𝕂=vΣF𝕂v𝕂subscriptproduct𝑣subscriptΣ𝐹subscript𝕂𝑣{\mathbb{K}}=\prod_{v\in\Sigma_{F}}{\mathbb{K}}_{v} viewed as a subgroup of the adelization G𝔸=vΣFGvsubscript𝐺𝔸superscriptsubscriptproduct𝑣subscriptΣ𝐹subscript𝐺𝑣G_{\mathbb{A}}=\prod_{v\in\Sigma_{F}}^{\prime}G_{v}. Given a non-zero ideal 𝔫𝔬𝔫𝔬{\mathfrak{n}}\subset\mathfrak{o} and an even weight l=(lv)vΣ(2)Σ𝑙subscriptsubscript𝑙𝑣𝑣subscriptΣsuperscript2subscriptΣl=(l_{v})_{v\in\Sigma_{\infty}}\in(2{\mathbb{N}})^{\Sigma_{\infty}}, let Πcus(l,𝔫)subscriptΠcus𝑙𝔫\Pi_{\rm{cus}}(l,{\mathfrak{n}}) be the set of all those irreducible cuspidal automorphic representations πvπv\pi\cong\otimes_{v}^{\prime}\pi_{v} of Z𝔸\G𝔸\subscript𝑍𝔸subscript𝐺𝔸Z_{\mathbb{A}}\backslash G_{\mathbb{A}} such that πvsubscript𝜋𝑣\pi_{v} is a discrete series representation of Zv\Gv\subscript𝑍𝑣subscript𝐺𝑣Z_{v}\backslash G_{v} of weight lvsubscript𝑙𝑣l_{v} for all vΣ𝑣subscriptΣv\in\Sigma_{\infty} and πvsubscript𝜋𝑣\pi_{v} has a non-zero vector invariant by the group 𝕂0(𝔫𝔬v)={[abcd]𝕂v|c𝔫𝔬v}subscript𝕂0𝔫subscript𝔬𝑣conditional-setdelimited-[]𝑎𝑏𝑐𝑑subscript𝕂𝑣𝑐𝔫subscript𝔬𝑣{\mathbb{K}}_{0}({\mathfrak{n}}\mathfrak{o}_{v})=\{\left[\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right]\in{\mathbb{K}}_{v}|\,c\in{\mathfrak{n}}\mathfrak{o}_{v}\,\} for all vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. Let vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} for a while. Set 𝔛v=/4πi(logqv)1subscript𝔛𝑣4𝜋𝑖superscriptsubscript𝑞𝑣1{\mathfrak{X}}_{v}={\mathbb{C}}/4\pi i(\log q_{v})^{-1}{\mathbb{Z}}. For a 𝕂vsubscript𝕂𝑣{\mathbb{K}}_{v}-spherical irreducible representation πvsubscript𝜋𝑣\pi_{v} of Gvsubscript𝐺𝑣G_{v} with trivial central character, we define

(1.1) Q(πv)=xv(π)/(qv1/2+qv1/2),xv(π)=av+av1,av=qvνv(π)/2formulae-sequence𝑄subscript𝜋𝑣subscript𝑥𝑣𝜋superscriptsubscript𝑞𝑣12superscriptsubscript𝑞𝑣12formulae-sequencesubscript𝑥𝑣𝜋subscript𝑎𝑣superscriptsubscript𝑎𝑣1subscript𝑎𝑣superscriptsubscript𝑞𝑣subscript𝜈𝑣𝜋2\displaystyle Q(\pi_{v})=x_{v}(\pi)/(q_{v}^{1/2}+q_{v}^{-1/2}),\quad x_{v}(\pi)=a_{v}+a_{v}^{-1},\quad a_{v}=q_{v}^{-\nu_{v}(\pi)/2}

with (av,av1)subscript𝑎𝑣superscriptsubscript𝑎𝑣1(a_{v},a_{v}^{-1}) the Satake parameter of πvsubscript𝜋𝑣\pi_{v}; note that the exponent νv(π)𝔛vsubscript𝜈𝑣𝜋subscript𝔛𝑣\nu_{v}(\pi)\in{\mathfrak{X}}_{v} is determined only up to sign. Let I(||vs)I(|\,|_{v}^{s}) denote the normalized induced representation IndBvGv(||vs||vs){\rm Ind}_{B_{v}}^{G_{v}}(|\,|_{v}^{s}\boxtimes|\,|_{v}^{-s}). Then πvI(||vνv(π)/2)\pi_{v}\cong I(|\,|_{v}^{\nu_{v}(\pi)/2}) and Q(πv)𝑄subscript𝜋𝑣Q(\pi_{v})\in{\mathbb{R}} if πvsubscript𝜋𝑣\pi_{v} is generic and unitarizable. For δFv×𝛿superscriptsubscript𝐹𝑣\delta\in F_{v}^{\times}, let εδsubscript𝜀𝛿\varepsilon_{\delta} denote the real-valued character of Fv×superscriptsubscript𝐹𝑣F_{v}^{\times} corresponding to the extension Fv(δ)/Fvsubscript𝐹𝑣𝛿subscript𝐹𝑣F_{v}(\sqrt{\delta})/F_{v} by local class field theory; it depends only on the coset δ(Fv×)2𝛿superscriptsuperscriptsubscript𝐹𝑣2\delta(F_{v}^{\times})^{2}. For z𝑧z\in{\mathbb{C}} and s𝑠s\in{\mathbb{C}} (such that Re(s)>(|Re(z)|1)/2Re𝑠Re𝑧12\operatorname{Re}(s)>(|\operatorname{Re}(z)|-1)/2), we define complex-valued functions 𝒪v,ϵδ,(z)superscriptsubscript𝒪𝑣italic-ϵ𝛿𝑧{\mathcal{O}}_{v,\epsilon}^{\delta,(z)} (ϵ=0,1italic-ϵ01\epsilon=0,1) and 𝒮vδ,(z)superscriptsubscript𝒮𝑣𝛿𝑧{\mathcal{S}}_{v}^{\delta,(z)} on Fv×superscriptsubscript𝐹𝑣F_{v}^{\times} as

𝒪v,ϵδ,(z)(a)superscriptsubscript𝒪𝑣italic-ϵ𝛿𝑧𝑎\displaystyle{\mathcal{O}}_{v,\epsilon}^{\delta,(z)}(a) =ζFv(z)LFv(z+12,εδ)(1+qvz+121+qv)ϵ|a|vz+14+ζFv(z)LFv(z+12,εδ)(1+qvz+121+qv)ϵ|a|vz+14absentsubscript𝜁subscript𝐹𝑣𝑧subscript𝐿subscript𝐹𝑣𝑧12subscript𝜀𝛿superscript1superscriptsubscript𝑞𝑣𝑧121subscript𝑞𝑣italic-ϵsuperscriptsubscript𝑎𝑣𝑧14subscript𝜁subscript𝐹𝑣𝑧subscript𝐿subscript𝐹𝑣𝑧12subscript𝜀𝛿superscript1superscriptsubscript𝑞𝑣𝑧121subscript𝑞𝑣italic-ϵsuperscriptsubscript𝑎𝑣𝑧14\displaystyle=\frac{\zeta_{F_{v}}(-z)}{L_{F_{v}}\left(\tfrac{-z+1}{2},\varepsilon_{\delta}\right)}\biggl{(}\frac{1+q_{v}^{\frac{z+1}{2}}}{1+q_{v}}\biggr{)}^{\epsilon}|a|_{v}^{\frac{-z+1}{4}}+\frac{\zeta_{F_{v}}(z)}{L_{F_{v}}\left(\tfrac{z+1}{2},\varepsilon_{\delta}\right)}\biggl{(}\frac{1+q_{v}^{\frac{-z+1}{2}}}{1+q_{v}}\biggr{)}^{\epsilon}|a|_{v}^{\frac{z+1}{4}}

and

(1.2) 𝒮vδ,(z)(s;a)superscriptsubscript𝒮𝑣𝛿𝑧𝑠𝑎\displaystyle{\mathcal{S}}_{v}^{\delta,(z)}(s;a) =qvs+12ζFv(s+z+12)ζFv(s+z+12)LFv(s+1,εδ)|a|vs+12,(|a|v1),absentsuperscriptsubscript𝑞𝑣𝑠12subscript𝜁subscript𝐹𝑣𝑠𝑧12subscript𝜁subscript𝐹𝑣𝑠𝑧12subscript𝐿subscript𝐹𝑣𝑠1subscript𝜀𝛿superscriptsubscript𝑎𝑣𝑠12subscript𝑎𝑣1\displaystyle=-q_{v}^{-\frac{s+1}{2}}\frac{\zeta_{F_{v}}\left(s+\frac{z+1}{2}\right)\zeta_{F_{v}}\left(s+\tfrac{-z+1}{2}\right)}{L_{F_{v}}(s+1,\varepsilon_{\delta})}|a|_{v}^{\frac{s+1}{2}},\quad(|a|_{v}\leqslant 1),
𝒮vδ,(z)(s;a)superscriptsubscript𝒮𝑣𝛿𝑧𝑠𝑎\displaystyle{\mathcal{S}}_{v}^{\delta,(z)}(s;a) =qvs+12{ζFv(z)ζFv(s+z+12)LFv(z+12,εδ)|a|vz+14+ζFv(z)ζFv(s+z+12)LFv(z+12,εδ)|a|vz+14},(|a|v>1).absentsuperscriptsubscript𝑞𝑣𝑠12conditional-setsubscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣𝑠𝑧12subscript𝐿subscript𝐹𝑣𝑧12subscript𝜀𝛿evaluated-at𝑎𝑣𝑧14subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣𝑠𝑧12subscript𝐿subscript𝐹𝑣𝑧12subscript𝜀𝛿superscriptsubscript𝑎𝑣𝑧14subscript𝑎𝑣1\displaystyle=-q_{v}^{-\frac{s+1}{2}}\left\{\frac{\zeta_{F_{v}}(-z)\zeta_{F_{v}}\left(s+\tfrac{z+1}{2}\right)}{L_{F_{v}}\left(\tfrac{-z+1}{2},\varepsilon_{\delta}\right)}\left|a\right|_{v}^{\frac{-z+1}{4}}+\frac{\zeta_{F_{v}}(z)\zeta_{F_{v}}\left(s+\tfrac{-z+1}{2}\right)}{L_{F_{v}}\left(\tfrac{z+1}{2},\varepsilon_{\delta}\right)}\left|a\right|_{v}^{\frac{z+1}{4}}\right\},\quad(|a|_{v}>1).

We remark that, when viewed as meromorphic functions in z𝑧z, the singularities of these functions at qvz/2=1superscriptsubscript𝑞𝑣𝑧21q_{v}^{z/2}=1 are removable. A computation reveals

(1.3) 𝒪v,0δ,(z)(a)=1,𝒪v,1δ,(z)(a)=11+qv{2(v splits in F(δ)/F),1(v ramifies in F(δ)/F),0(v remains prime in F(δ)/F) for all a𝔬v×.formulae-sequencesuperscriptsubscript𝒪𝑣0𝛿𝑧𝑎1superscriptsubscript𝒪𝑣1𝛿𝑧𝑎11subscript𝑞𝑣cases2v splits in F(δ)/Fotherwise1v ramifies in F(δ)/Fotherwise0v remains prime in F(δ)/Fotherwise for all a𝔬v×.\displaystyle{\mathcal{O}}_{v,0}^{\delta,(z)}(a)=1,\quad{\mathcal{O}}_{v,1}^{\delta,(z)}(a)=\tfrac{1}{1+q_{v}}\begin{cases}2\quad(\text{$v$ splits in $F(\sqrt{\delta})/F$}),\\ 1\quad(\text{$v$ ramifies in $F(\sqrt{\delta})/F$}),\\ 0\quad(\text{$v$ remains prime in $F(\sqrt{\delta})/F$})\end{cases}\quad\text{ for all $a\in\mathfrak{o}_{v}^{\times}$.}

For a finite subset SΣfin𝑆subscriptΣfinS\subset\Sigma_{\rm fin}, a square-free ideal 𝔫𝔫{\mathfrak{n}} such that SS(𝔫)=𝑆𝑆𝔫S\cap S({\mathfrak{n}})=\varnothing, an element ΔF×Δsuperscript𝐹\Delta\in F^{\times}, and a non-zero fractional ideal 𝔞F𝔞𝐹{\mathfrak{a}}\subset F, the product

𝐁𝔫(z)(𝐬|Δ;𝔞)=vΣfin(SS(𝔫))𝒪0,vΔ,(z)(av)vS(𝔫)𝒪1,vΔ,(z)(av)vS𝒮vΔ,(z)(sv,av),superscriptsubscript𝐁𝔫𝑧conditional𝐬Δ𝔞subscriptproduct𝑣subscriptΣfin𝑆𝑆𝔫superscriptsubscript𝒪0𝑣Δ𝑧subscript𝑎𝑣subscriptproduct𝑣𝑆𝔫superscriptsubscript𝒪1𝑣Δ𝑧subscript𝑎𝑣subscriptproduct𝑣𝑆superscriptsubscript𝒮𝑣Δ𝑧subscript𝑠𝑣subscript𝑎𝑣\displaystyle{\bf B}_{{\mathfrak{n}}}^{(z)}({\mathbf{s}}|\Delta;{\mathfrak{a}})=\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}{\mathcal{O}}_{0,v}^{\Delta,(z)}(a_{v})\prod_{v\in S({\mathfrak{n}})}{\mathcal{O}}_{1,v}^{\Delta,(z)}(a_{v})\prod_{v\in S}{\mathcal{S}}_{v}^{\Delta,(z)}(s_{v},a_{v}),\qquad
(𝐬𝔛S,minvSRe(sv)>(|Re(z)|1)/2)formulae-sequence𝐬subscript𝔛𝑆subscript𝑣𝑆Resubscript𝑠𝑣Re𝑧12\displaystyle\quad(\ {\mathbf{s}}\in{\mathfrak{X}}_{S},\quad\min_{v\in S}\operatorname{Re}(s_{v})>(|\operatorname{Re}(z)|-1)/2\ )

is well-defined due to (1.3), where (av)𝔸fin×subscript𝑎𝑣superscriptsubscript𝔸fin(a_{v})\in{\mathbb{A}}_{{\rm fin}}^{\times} is the idele corresponding to 𝔞𝔞{\mathfrak{a}} and 𝔛S=vS𝔛vsubscript𝔛𝑆subscriptproduct𝑣𝑆subscript𝔛𝑣{\mathfrak{X}}_{S}=\prod_{v\in S}{\mathfrak{X}}_{v}. Let vΣ𝑣subscriptΣv\in\Sigma_{\infty}. We define a complex-valued function 𝒪v±,(z)superscriptsubscript𝒪𝑣plus-or-minus𝑧{\mathcal{O}}_{v}^{\pm,(z)} on Fv×superscriptsubscript𝐹𝑣F_{v}^{\times} as

𝒪v+,(z)(a)superscriptsubscript𝒪𝑣𝑧𝑎\displaystyle{\mathcal{O}}_{v}^{+,(z)}(a) =2πΓ(lv)Γ(lv+z12)Γ(lv+z12)Γ(1+z2)Γ(1z2)δ(|a|>1)(a21)1/2𝔓z121lv(|a|),absent2𝜋Γsubscript𝑙𝑣Γsubscript𝑙𝑣𝑧12Γsubscript𝑙𝑣𝑧12subscriptΓ1𝑧2subscriptΓ1𝑧2𝛿𝑎1superscriptsuperscript𝑎2112subscriptsuperscript𝔓1subscript𝑙𝑣𝑧12𝑎\displaystyle=\frac{2\pi}{\Gamma(l_{v})}\frac{\Gamma\left(l_{v}+\tfrac{z-1}{2}\right)\Gamma\left(l_{v}+\tfrac{-z-1}{2}\right)}{\Gamma_{\mathbb{R}}\left(\tfrac{1+z}{2}\right)\Gamma_{\mathbb{R}}\left(\tfrac{1-z}{2}\right)}\delta(|a|>1)(a^{2}-1)^{1/2}{\mathfrak{P}}^{1-l_{v}}_{\frac{z-1}{2}}(|a|),\quad
𝒪v,(z)(a)superscriptsubscript𝒪𝑣𝑧𝑎\displaystyle{\mathcal{O}}_{v}^{-,(z)}(a) =πiΓ(lv)Γ(lv+z12)Γ(lv+z12)sgn(a)(1+a2)1/2{𝔓z121lv(ia)𝔓z121lv(ia)},absent𝜋𝑖Γsubscript𝑙𝑣Γsubscript𝑙𝑣𝑧12Γsubscript𝑙𝑣𝑧12sgn𝑎superscript1superscript𝑎212subscriptsuperscript𝔓1subscript𝑙𝑣𝑧12𝑖𝑎subscriptsuperscript𝔓1subscript𝑙𝑣𝑧12𝑖𝑎\displaystyle=\frac{\pi i}{\Gamma(l_{v})}\Gamma\left(l_{v}+\tfrac{z-1}{2}\right)\Gamma\left(l_{v}+\tfrac{-z-1}{2}\right)\,{\rm sgn}(a)\,(1+a^{2})^{1/2}\{{\mathfrak{P}}^{1-l_{v}}_{\frac{z-1}{2}}(ia)-{\mathfrak{P}}^{1-l_{v}}_{\frac{z-1}{2}}(-ia)\},

where 𝔓νμ(x)superscriptsubscript𝔓𝜈𝜇𝑥{\mathfrak{P}}_{\nu}^{\mu}(x) is the Legendre function of the 1st kind which is defined for points x𝑥x\in{\mathbb{C}} outside the interval (,+1]1(-\infty,+1] of the real axis ([16, §4.1]).

Suppose 𝔫𝔫{\mathfrak{n}} is square-free from now on. For πvπvΠcus(l,𝔫)\pi\cong\otimes_{v}\pi_{v}\in\Pi_{\rm cus}(l,{\mathfrak{n}}) and a finite set SΣfin𝑆subscriptΣfinS\subset\Sigma_{\rm fin} disjoint from S(𝔫)𝑆𝔫S({\mathfrak{n}}), set νS(π)={νv(π)}vSsubscript𝜈𝑆𝜋subscriptsubscript𝜈𝑣𝜋𝑣𝑆\nu_{S}(\pi)=\{\nu_{v}(\pi)\}_{v\in S}, an element of 𝔛S/{±1}S=vS(𝔛v/{±1})subscript𝔛𝑆superscriptplus-or-minus1𝑆subscriptproduct𝑣𝑆subscript𝔛𝑣plus-or-minus1{\mathfrak{X}}_{S}/\{\pm 1\}^{S}=\prod_{v\in S}({\mathfrak{X}}_{v}/\{\pm 1\}). We set

W𝔫(z)(π)subscriptsuperscript𝑊𝑧𝔫𝜋\displaystyle W^{(z)}_{{\mathfrak{n}}}(\pi) =N(𝔫𝔣π1)(1z)/2vS(𝔫𝔣π1){1+Q(Iv(||vz/2))Q(πv)21Q(πv)2},\displaystyle={\operatorname{N}}({\mathfrak{n}}{\mathfrak{f}}_{\pi}^{-1})^{(1-z)/2}\prod_{v\in S({\mathfrak{n}}{\mathfrak{f}}_{\pi}^{-1})}\biggl{\{}1+\frac{Q(I_{v}(|\,|_{v}^{z/2}))-Q(\pi_{v})^{2}}{1-Q(\pi_{v})^{2}}\biggr{\}},

where 𝔣πsubscript𝔣𝜋{\mathfrak{f}}_{\pi} denotes the conductor of π𝜋\pi. Since 1<Q(πv)<11𝑄subscript𝜋𝑣1-1<Q(\pi_{v})<1 by the unitarity of πvsubscript𝜋𝑣\pi_{v}, we have W𝔫(z)(π)0subscriptsuperscript𝑊𝑧𝔫𝜋0W^{(z)}_{\mathfrak{n}}(\pi)\geqslant 0 when z𝑧z is a non-negative real number. Let L(s,π;Ad)𝐿𝑠𝜋AdL(s,\pi;{\operatorname{Ad}}) be the symmetric square L𝐿L-function of πΠcus(l,𝔫)𝜋subscriptΠcus𝑙𝔫\pi\in\Pi_{\rm cus}(l,{\mathfrak{n}}), which on Re(s)>1Re𝑠1\operatorname{Re}(s)>1 is defined by the Euler product of (1av2qvs)1(1qvs)1(1av2qvs)1superscript1superscriptsubscript𝑎𝑣2superscriptsubscript𝑞𝑣𝑠1superscript1superscriptsubscript𝑞𝑣𝑠1superscript1superscriptsubscript𝑎𝑣2superscriptsubscript𝑞𝑣𝑠1(1-a_{v}^{2}q_{v}^{-s})^{-1}(1-q_{v}^{-s})^{-1}(1-a_{v}^{-2}q_{v}^{-s})^{-1} over vΣfinS(𝔣π)𝑣subscriptΣfin𝑆subscript𝔣𝜋v\in\Sigma_{\rm fin}-S({\mathfrak{f}}_{\pi}) completed by the gamma factor Γ(s+1)Γ(s+lv1)subscriptΓ𝑠1subscriptΓ𝑠subscript𝑙𝑣1\Gamma_{\mathbb{R}}(s+1)\Gamma_{{\mathbb{C}}}(s+l_{v}-1) over vΣ𝑣subscriptΣv\in\Sigma_{\infty} and by finitely many factors (1qvs1)1superscript1superscriptsubscript𝑞𝑣𝑠11(1-q_{v}^{-s-1})^{-1} over vS(𝔣π)𝑣𝑆subscript𝔣𝜋v\in S({\mathfrak{f}}_{\pi}); it is known to be entire on {\mathbb{C}} and is identified with the standard L𝐿L-function of an automorphic representation Ad(π)Ad𝜋{\operatorname{Ad}}(\pi) of GL(3,𝔸)GL3𝔸{\operatorname{GL}}(3,{\mathbb{A}}) ([6]). We have the functional equation L(s,π;Ad)=DF3(1/2s)N(𝔣π)2(1/2s)L(1s,π;Ad)L(s,\pi;{\operatorname{Ad}})=D_{F}^{3(1/2-s)}\,{\operatorname{N}}({\mathfrak{f}}_{\pi})^{2(1/2-s)}\,L(1-s,\pi;{\operatorname{Ad}}). In particular, the sign of the functional equation is plus. Let I^cusp0(𝐬,z)superscriptsubscript^𝐼cusp0𝐬𝑧\hat{I}_{\rm{cusp}}^{0}({\mathbf{s}},z) be a meromorphic function on 𝔛S×subscript𝔛𝑆{\mathfrak{X}}_{S}\times{\mathbb{C}} defined as

(1.4) I^cusp0(𝐬,z)superscriptsubscript^𝐼cusp0𝐬𝑧\displaystyle\hat{I}_{\rm cusp}^{0}({\mathbf{s}},z)
=\displaystyle= C(l,𝔫)πΠcus(l,𝔫)21DFz1/2N(𝔫)(z1)/2W𝔫(z)(π)vS{(qv(1+νv(π))/2+qv(1νv(π))/2)(qv(1+sv)/2+qv(1sv)/2)}L(z+12,π;Ad)L(1,π;Ad),\displaystyle C(l,{\mathfrak{n}})\sum_{\pi\in\Pi_{\rm{cus}}(l,{\mathfrak{n}})}\frac{2^{-1}D_{F}^{z-1/2}{\operatorname{N}}({\mathfrak{n}})^{(z-1)/2}W_{{\mathfrak{n}}}^{(z)}(\pi)}{\prod_{v\in S}\{(q_{v}^{(1+\nu_{v}(\pi))/2}+q_{v}^{(1-\nu_{v}(\pi))/2})-(q_{v}^{(1+s_{v})/2}+q_{v}^{(1-s_{v})/2})\}}\frac{L(\frac{z+1}{2},\pi;{\operatorname{Ad}})}{L(1,\pi;{\operatorname{Ad}})},

where

(1.5) C(l,𝔫)=DF1{vΣ4πlv1}{vS(𝔫)(1+qv)1}.𝐶𝑙𝔫superscriptsubscript𝐷𝐹1subscriptproduct𝑣subscriptΣ4𝜋subscript𝑙𝑣1subscriptproduct𝑣𝑆𝔫superscript1subscript𝑞𝑣1\displaystyle C(l,{\mathfrak{n}})=D_{F}^{-1}\,\{\prod_{v\in\Sigma_{\infty}}\tfrac{4\pi}{l_{v}-1}\}\,\{\prod_{v\in S({\mathfrak{n}})}(1+q_{v})^{-1}\}.

For ΔF×(F×)2Δsuperscript𝐹superscriptsuperscript𝐹2\Delta\in F^{\times}-(F^{\times})^{2}, let L(s,εΔ)𝐿𝑠subscript𝜀ΔL(s,\varepsilon_{\Delta}) be the completed L𝐿L-function of the class field character εΔsubscript𝜀Δ\varepsilon_{\Delta} of the quadratic extension F(Δ)/F𝐹Δ𝐹F(\sqrt{\Delta})/F, 𝔡Δsubscript𝔡Δ{\mathfrak{d}}_{\Delta} the relative discriminant of F(Δ)/F𝐹Δ𝐹F(\sqrt{\Delta})/F, and 𝔣Δsubscript𝔣Δ{\mathfrak{f}}_{\Delta} a fractional ideal uniquely determined by Δ𝔬=𝔡Δ𝔣Δ2Δ𝔬subscript𝔡Δsuperscriptsubscript𝔣Δ2\Delta\mathfrak{o}={\mathfrak{d}}_{\Delta}{\mathfrak{f}}_{\Delta}^{2}.

Theorem 1.1.

Let l=(lv)vΣ(2)Σ𝑙subscriptsubscript𝑙𝑣𝑣subscriptΣsuperscript2subscriptΣl=(l_{v})_{v\in\Sigma_{\infty}}\in(2{\mathbb{N}})^{\Sigma_{\infty}} with lv4subscript𝑙𝑣4l_{v}\geqslant 4 for all vΣ𝑣subscriptΣv\in\Sigma_{\infty} and 𝔫𝔫{\mathfrak{n}} a non-zero square-free ideal of 𝔬𝔬\mathfrak{o}. Let S𝑆S be a finite subset of ΣfinsubscriptΣfin\Sigma_{\rm fin} such that SS(𝔫)=𝑆𝑆𝔫S\cap S({\mathfrak{n}})=\varnothing. We suppose 222 splits completely in F/𝐹F/{\mathbb{Q}} and |2|v=1subscript2𝑣1|2|_{v}=1 for all vSS(𝔫)𝑣𝑆𝑆𝔫v\in S\cup S({\mathfrak{n}}). Then, for 𝐬𝔛S𝐬subscript𝔛𝑆{\mathbf{s}}\in{\mathfrak{X}}_{S} and z𝑧z\in\mathbb{C} such that minvSRe(sv)>2minvΣlv1subscript𝑣𝑆Resubscript𝑠𝑣2subscript𝑣subscriptΣsubscript𝑙𝑣1\min_{v\in S}\operatorname{Re}(s_{v})>2\min_{v\in\Sigma_{\infty}}l_{v}-1 and |Re(z)|<minvΣlv3Re𝑧subscript𝑣subscriptΣsubscript𝑙𝑣3|\operatorname{Re}(z)|<\min_{v\in\Sigma_{\infty}}l_{v}-3, we have the identity

I^cusp0(𝐬,z)=subscriptsuperscript^𝐼0cusp𝐬𝑧absent\displaystyle\hat{I}^{0}_{\rm cusp}({\mathbf{s}},z)= DFz4{J^unip0(𝐬,z)+J^unip0(𝐬,z)}+J^hyp0(𝐬,z)+J^ell0(𝐬,z),superscriptsubscript𝐷𝐹𝑧4superscriptsubscript^𝐽unip0𝐬𝑧superscriptsubscript^𝐽unip0𝐬𝑧superscriptsubscript^𝐽hyp0𝐬𝑧superscriptsubscript^𝐽ell0𝐬𝑧\displaystyle D_{F}^{\frac{z}{4}}\{\hat{J}_{\rm unip}^{0}({\mathbf{s}},z)+\hat{J}_{\rm unip}^{0}({\mathbf{s}},-z)\}+\hat{J}_{\rm hyp}^{0}({\mathbf{s}},z)+\hat{J}_{\rm ell}^{0}({\mathbf{s}},z),

where the terms on the right-hand side are described as follows. The first term is defined as

J^unip0(𝐬,z)superscriptsubscript^𝐽unip0𝐬𝑧\displaystyle\hat{J}_{\rm unip}^{0}({\mathbf{s}},z) =DFz+24ζF(z)vSqv(sv+1)/21qvsv(z+1)/2vS(𝔫)1+qvz+121+qvvΣ21zπ3z4Γ(lv+z12)Γ(z+14)Γ(lv).absentsuperscriptsubscript𝐷𝐹𝑧24subscript𝜁𝐹𝑧subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣subscript𝑠𝑣121superscriptsubscript𝑞𝑣subscript𝑠𝑣𝑧12subscriptproduct𝑣𝑆𝔫1superscriptsubscript𝑞𝑣𝑧121subscript𝑞𝑣subscriptproduct𝑣subscriptΣsuperscript21𝑧superscript𝜋3𝑧4Γsubscript𝑙𝑣𝑧12Γ𝑧14Γsubscript𝑙𝑣\displaystyle=D_{F}^{-\frac{z+2}{4}}\zeta_{F}(-z)\,\prod_{v\in S}\frac{-q_{v}^{-{(s_{v}+1)}/{2}}}{1-q_{v}^{-s_{v}-(z+1)/{2}}}\,\prod_{v\in S({\mathfrak{n}})}\frac{1+q_{v}^{\frac{z+1}{2}}}{1+q_{v}}\,\prod_{v\in\Sigma_{\infty}}2^{1-z}\pi^{\frac{3-z}{4}}\frac{\Gamma\left(l_{v}+\frac{z-1}{2}\right)}{\Gamma\left(\tfrac{z+1}{4}\right)\Gamma(l_{v})}.

The second is given by the absolutely convergent sum

J^hyp0(𝐬,z)superscriptsubscript^𝐽hyp0𝐬𝑧\displaystyle\hat{J}_{\rm hyp}^{0}({\mathbf{s}},z) =12DF1/2ζF(1z2)a𝔬(S)+×{1}𝐁𝔫(z)(𝐬|1;a(a1)2𝔬)vΣ𝒪v+,(z)((a+1)/(a1)),absent12superscriptsubscript𝐷𝐹12subscript𝜁𝐹1𝑧2subscript𝑎𝔬superscriptsubscript𝑆1superscriptsubscript𝐁𝔫𝑧conditional𝐬1𝑎superscript𝑎12𝔬subscriptproduct𝑣subscriptΣsuperscriptsubscript𝒪𝑣𝑧𝑎1𝑎1\displaystyle=\tfrac{1}{2}D_{F}^{-1/2}\zeta_{F}\left(\tfrac{1-z}{2}\right)\,\sum_{a\in\mathfrak{o}(S)_{+}^{\times}-\{1\}}{\bf B}_{{\mathfrak{n}}}^{(z)}({\mathbf{s}}|1;a(a-1)^{-2}\mathfrak{o})\,\prod_{v\in\Sigma_{\infty}}{\mathcal{O}}_{v}^{+,(z)}((a+1)/(a-1)),

where 𝔬(S)+×𝔬superscriptsubscript𝑆\mathfrak{o}(S)_{+}^{\times} is the totally positive units of the S𝑆S-integer ring of F𝐹F. The third term is given by the absolutely convergent sum

J^ell0(𝐬,z)superscriptsubscript^𝐽ell0𝐬𝑧\displaystyle\hat{J}_{\rm ell}^{0}({\mathbf{s}},z) =12DFz12(t:n)FN(𝔡Δ)z+14L(z+12,εΔ)𝐁𝔫(z)(𝐬|Δ;n𝔣Δ2)vΣ𝒪vsgn(Δ(v)),(z)(t|Δ|v1/2),\displaystyle=\tfrac{1}{2}D_{F}^{\frac{z-1}{2}}\sum_{(t:n)_{F}}{\operatorname{N}}({\mathfrak{d}}_{\Delta})^{\frac{z+1}{4}}L\left(\tfrac{z+1}{2},\varepsilon_{\Delta}\right)\,{\bf B}_{{\mathfrak{n}}}^{(z)}({\mathbf{s}}|\Delta;n{\mathfrak{f}}_{\Delta}^{-2})\,\prod_{v\in\Sigma_{\infty}}{\mathcal{O}}_{v}^{{\operatorname{sgn}}(\Delta^{(v)}),(z)}(t|\Delta|_{v}^{-1/2}),

where (t:n)F(t:n)_{F} runs over different cosets {(ct,c2n)F2|cF×}conditional-set𝑐𝑡superscript𝑐2𝑛superscript𝐹2𝑐superscript𝐹\{(ct,c^{2}n)\in F^{2}|\,c\in F^{\times}\} such that Δ=t24nF×(F×)2Δsuperscript𝑡24𝑛superscript𝐹superscriptsuperscript𝐹2\Delta=t^{2}-4n\in F^{\times}-(F^{\times})^{2}, (t,n){(cvtv,cv2nv)|cvFv×,tv𝔬v,nv𝔬v×}𝑡𝑛conditional-setsubscript𝑐𝑣subscript𝑡𝑣superscriptsubscript𝑐𝑣2subscript𝑛𝑣formulae-sequencesubscript𝑐𝑣superscriptsubscript𝐹𝑣formulae-sequencesubscript𝑡𝑣subscript𝔬𝑣subscript𝑛𝑣superscriptsubscript𝔬𝑣(t,n)\in\{(c_{v}t_{v},c_{v}^{2}n_{v})|\,c_{v}\in F_{v}^{\times},\,t_{v}\in\mathfrak{o}_{v},\,n_{v}\in\mathfrak{o}_{v}^{\times}\} for all vΣfinS𝑣subscriptΣfin𝑆v\in\Sigma_{\rm fin}-S, and ordv(n𝔣Δ2)<0subscriptord𝑣𝑛superscriptsubscript𝔣Δ20\operatorname{ord}_{v}(n{\mathfrak{f}}_{\Delta}^{-2})<0 for all vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}) with εΔ,vsubscript𝜀Δ𝑣\varepsilon_{\Delta,v} being unramified and non-trivial.

As a corollary of Theorem 1.1, we have a generalization of [35], [18] and [29]. Let 𝒜vsubscript𝒜𝑣{\mathcal{A}}_{v} be the space of holomorphic functions on 𝔛vsubscript𝔛𝑣{\mathfrak{X}}_{v} such that α(sv)=α(sv)𝛼subscript𝑠𝑣𝛼subscript𝑠𝑣\alpha(-s_{v})=\alpha(s_{v}). Let dμv(sv)𝑑subscript𝜇𝑣subscript𝑠𝑣{{d}}\mu_{v}(s_{v}) denote the holomorphic 111-form 21(logqv)(qv(1+sv)/2qv(1sv)/2)dsvsuperscript21subscript𝑞𝑣superscriptsubscript𝑞𝑣1subscript𝑠𝑣2superscriptsubscript𝑞𝑣1subscript𝑠𝑣2𝑑subscript𝑠𝑣2^{-1}(\log q_{v})(q_{v}^{(1+s_{v})/2}-q_{v}^{(1-s_{v})/2})\,{{d}}s_{v} on 𝔛vsubscript𝔛𝑣{\mathfrak{X}}_{v}, and let Lv(c)subscript𝐿𝑣𝑐L_{v}(c) be the contour yc+iy(c,y,|y|2π(logqv)1)y\mapsto c+iy\,(c,y\in{\mathbb{R}},\,|y|\leqslant{2\pi}{(\log q_{v})}^{-1}). For α𝒜v𝛼subscript𝒜𝑣\alpha\in{\mathcal{A}}_{v}, set

𝒮^vδ,(z)(α;a)=12πiLv(c)𝒮vδ,(z)(s;a)α(s)𝑑μv(s).superscriptsubscript^𝒮𝑣𝛿𝑧𝛼𝑎12𝜋𝑖subscriptsubscript𝐿𝑣𝑐superscriptsubscript𝒮𝑣𝛿𝑧𝑠𝑎𝛼𝑠differential-dsubscript𝜇𝑣𝑠\hat{\mathcal{S}}_{v}^{\delta,(z)}(\alpha;a)=\tfrac{1}{2\pi i}\int_{L_{v}(c)}{\mathcal{S}}_{v}^{\delta,(z)}(s;a)\,\alpha(s)\,{{d}}\mu_{v}(s).

For a finite subset SΣfin𝑆subscriptΣfinS\subset\Sigma_{\rm fin}, let α(𝐬)=vSαv(sv)\alpha({\mathbf{s}})=\otimes_{v\in S}\alpha_{v}(s_{v}) be a pure tensor of 𝒜S=vS𝒜vsubscript𝒜𝑆subscripttensor-product𝑣𝑆subscript𝒜𝑣{\mathcal{A}}_{S}=\bigotimes_{v\in S}{\mathcal{A}}_{v}, viewed as a function on 𝔛Ssubscript𝔛𝑆{\mathfrak{X}}_{S}. For a square-free ideal 𝔫𝔫{\mathfrak{n}} such that SS(𝔫)=𝑆𝑆𝔫S\cap S({\mathfrak{n}})=\varnothing, an element ΔF×Δsuperscript𝐹\Delta\in F^{\times}, and a non-zero fractional ideal 𝔞F𝔞𝐹{\mathfrak{a}}\subset F and z𝑧z\in\mathbb{C}, set

𝐁𝔫(z)(α|Δ;𝔞)=vΣfin(SS(𝔫))𝒪0,vΔ,(z)(av)vS(𝔫)𝒪1,vΔ,(z)(av)vS𝒮^vΔ,(z)(αv,av),superscriptsubscript𝐁𝔫𝑧conditional𝛼Δ𝔞subscriptproduct𝑣subscriptΣfin𝑆𝑆𝔫superscriptsubscript𝒪0𝑣Δ𝑧subscript𝑎𝑣subscriptproduct𝑣𝑆𝔫superscriptsubscript𝒪1𝑣Δ𝑧subscript𝑎𝑣subscriptproduct𝑣𝑆superscriptsubscript^𝒮𝑣Δ𝑧subscript𝛼𝑣subscript𝑎𝑣{\bf B}_{{\mathfrak{n}}}^{(z)}(\alpha|\Delta;{\mathfrak{a}})=\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}{\mathcal{O}}_{0,v}^{\Delta,(z)}(a_{v})\prod_{v\in S({\mathfrak{n}})}{\mathcal{O}}_{1,v}^{\Delta,(z)}(a_{v})\prod_{v\in S}\hat{\mathcal{S}}_{v}^{\Delta,(z)}(\alpha_{v},a_{v}),
Υ(z)(α)=vS12πiLv(cv)qv(sv+1)/21qvsv(z+1)/2αv(sv)𝑑μv(sv)superscriptΥ𝑧𝛼subscriptproduct𝑣𝑆12𝜋𝑖subscriptsubscript𝐿𝑣subscript𝑐𝑣superscriptsubscript𝑞𝑣subscript𝑠𝑣121superscriptsubscript𝑞𝑣subscript𝑠𝑣𝑧12subscript𝛼𝑣subscript𝑠𝑣differential-dsubscript𝜇𝑣subscript𝑠𝑣\Upsilon^{(z)}(\alpha)=\prod_{v\in S}\tfrac{1}{2\pi i}\int_{L_{v}(c_{v})}\frac{-q_{v}^{-{(s_{v}+1)}/{2}}}{1-q_{v}^{-s_{v}-(z+1)/{2}}}\alpha_{v}(s_{v}){{d}}\mu_{v}(s_{v})

and

(1.6) 𝕀cusp0(𝔫|α,z)=subscriptsuperscript𝕀0cuspconditional𝔫𝛼𝑧absent\displaystyle\mathbb{I}^{0}_{\rm cusp}({\mathfrak{n}}|\alpha,z)= 21DFz1/2N(𝔫)(z1)/2πΠcus(l,𝔫)W𝔫(z)(π)L(z+12,π;Ad)L(1,π;Ad)α(νS(π)).\displaystyle 2^{-1}D_{F}^{z-1/2}{\operatorname{N}}({\mathfrak{n}})^{(z-1)/2}\sum_{\pi\in\Pi_{\rm cus}(l,{\mathfrak{n}})}W_{{\mathfrak{n}}}^{(z)}(\pi)\,\frac{L\left(\tfrac{z+1}{2},\pi;{\operatorname{Ad}}\right)}{L(1,\pi;{\operatorname{Ad}})}\,\alpha(\nu_{S}(\pi)).
Corollary 1.2.

Let l𝑙l, 𝔫𝔫{\mathfrak{n}}, S𝑆S be as in Theorem 1.1. We suppose 222 splits completely in F/𝐹F/{\mathbb{Q}} and |2|v=1subscript2𝑣1|2|_{v}=1 for all vSS(𝔫)𝑣𝑆𝑆𝔫v\in S\cup S({\mathfrak{n}}). Then for α𝒜S𝛼subscript𝒜𝑆\alpha\in{\mathcal{A}}_{S} and z𝑧z\in\mathbb{C} such that |Re(z)|<minvΣlv3Re𝑧subscript𝑣subscriptΣsubscript𝑙𝑣3|\operatorname{Re}(z)|<\min_{v\in\Sigma_{\infty}}l_{v}-3, we have the identity

(1)#SC(l,𝔫)𝕀cusp0(𝔫|α,z)=superscript1#𝑆𝐶𝑙𝔫subscriptsuperscript𝕀0cuspconditional𝔫𝛼𝑧absent\displaystyle(-1)^{\#S}C(l,{\mathfrak{n}})\,\mathbb{I}^{0}_{\rm cusp}({\mathfrak{n}}|\alpha,z)= DFz4{𝕁unip0(𝔫|α,z)+𝕁unip0(𝔫|α,z)}+𝕁hyp0(𝔫|α,z)+𝕁ell0(𝔫|α,z),superscriptsubscript𝐷𝐹𝑧4superscriptsubscript𝕁unip0conditional𝔫𝛼𝑧superscriptsubscript𝕁unip0conditional𝔫𝛼𝑧superscriptsubscript𝕁hyp0conditional𝔫𝛼𝑧superscriptsubscript𝕁ell0conditional𝔫𝛼𝑧\displaystyle D_{F}^{\frac{z}{4}}\{{\mathbb{J}}_{\rm unip}^{0}({\mathfrak{n}}|\alpha,z)+{\mathbb{J}}_{\rm unip}^{0}({\mathfrak{n}}|\alpha,-z)\}+{\mathbb{J}}_{\rm hyp}^{0}({\mathfrak{n}}|\alpha,z)+{\mathbb{J}}_{\rm ell}^{0}({\mathfrak{n}}|\alpha,z),

where C(l,𝔫)𝐶𝑙𝔫C(l,{\mathfrak{n}}) is the number defined as (1.5) and the terms on the right-hand side are described as follows. The first term is defined as

𝕁unip0(𝔫|α,z)superscriptsubscript𝕁unip0conditional𝔫𝛼𝑧\displaystyle{\mathbb{J}}_{\rm unip}^{0}({\mathfrak{n}}|\alpha,z) =DFz+24ζF(z)Υ(z)(α)vS(𝔫)1+qvz+121+qvvΣ21zπ3z4Γ(lv+z12)Γ(z+14)Γ(lv).absentsuperscriptsubscript𝐷𝐹𝑧24subscript𝜁𝐹𝑧superscriptΥ𝑧𝛼subscriptproduct𝑣𝑆𝔫1superscriptsubscript𝑞𝑣𝑧121subscript𝑞𝑣subscriptproduct𝑣subscriptΣsuperscript21𝑧superscript𝜋3𝑧4Γsubscript𝑙𝑣𝑧12Γ𝑧14Γsubscript𝑙𝑣\displaystyle=D_{F}^{-\frac{z+2}{4}}\zeta_{F}(-z)\,\Upsilon^{(z)}(\alpha)\,\prod_{v\in S({\mathfrak{n}})}\frac{1+q_{v}^{\frac{z+1}{2}}}{1+q_{v}}\,\prod_{v\in\Sigma_{\infty}}2^{1-z}\pi^{\frac{3-z}{4}}\frac{\Gamma\left(l_{v}+\frac{z-1}{2}\right)}{\Gamma\left(\tfrac{z+1}{4}\right)\Gamma(l_{v})}.

The second is given by the absolutely convergent sum

𝕁hyp0(𝔫|α,z)superscriptsubscript𝕁hyp0conditional𝔫𝛼𝑧\displaystyle{\mathbb{J}}_{\rm hyp}^{0}({\mathfrak{n}}|\alpha,z) =12DF1/2ζF(1z2)a𝔬(S)+×{1}𝐁𝔫(z)(α|1;a(a1)2𝔬)vΣ𝒪v+,(z)((a+1)/(a1)).absent12superscriptsubscript𝐷𝐹12subscript𝜁𝐹1𝑧2subscript𝑎𝔬superscriptsubscript𝑆1superscriptsubscript𝐁𝔫𝑧conditional𝛼1𝑎superscript𝑎12𝔬subscriptproduct𝑣subscriptΣsuperscriptsubscript𝒪𝑣𝑧𝑎1𝑎1\displaystyle=\tfrac{1}{2}D_{F}^{-1/2}\zeta_{F}\left(\tfrac{1-z}{2}\right)\,\sum_{a\in\mathfrak{o}(S)_{+}^{\times}-\{1\}}{\bf B}_{{\mathfrak{n}}}^{(z)}(\alpha|1;a(a-1)^{-2}\mathfrak{o})\,\prod_{v\in\Sigma_{\infty}}{\mathcal{O}}_{v}^{+,(z)}((a+1)/(a-1)).

The third term is given by the absolutely convergent sum

𝕁ell0(𝔫|α,z)superscriptsubscript𝕁ell0conditional𝔫𝛼𝑧\displaystyle{\mathbb{J}}_{\rm ell}^{0}({\mathfrak{n}}|\alpha,z) =12DFz12(t:n)FN(𝔡Δ)z+14L(z+12,εΔ)𝐁𝔫(z)(α|Δ;n𝔣Δ2)vΣ𝒪vsgn(Δ(v)),(z)(t|Δ|v1/2),\displaystyle=\tfrac{1}{2}D_{F}^{\frac{z-1}{2}}\sum_{(t:n)_{F}}{\operatorname{N}}({\mathfrak{d}}_{\Delta})^{\frac{z+1}{4}}L\left(\tfrac{z+1}{2},\varepsilon_{\Delta}\right)\,{\bf B}_{{\mathfrak{n}}}^{(z)}(\alpha|\Delta;n{\mathfrak{f}}_{\Delta}^{-2})\,\prod_{v\in\Sigma_{\infty}}{\mathcal{O}}_{v}^{{\operatorname{sgn}}(\Delta^{(v)}),(z)}(t|\Delta|_{v}^{-1/2}),

where Δ=t24nΔsuperscript𝑡24𝑛\Delta=t^{2}-4n and (t:n)F(t:n)_{F} runs over the same set as in Theorem 1.1.

Corollary 1.2 includes known formulas as a special case. Indeed, it is confirmed that [35, Theorem 1], [18, (4.6) and (5.9)], and [29, Proposition 2] are all recovered from Corollary 1.2. Moreover, we can deduce an explicit trace formula of Hecke operators in a more general setting than in these works.

1.2.1. A limit formula

For a real number z[0,1]𝑧01z\in[0,1] and a square-free ideal 𝔫𝔬𝔫𝔬{\mathfrak{n}}\subset\mathfrak{o}, let us define a discrete Radon measure on ΩS=[2,+2]SsubscriptΩ𝑆superscript22𝑆\Omega_{S}=[-2,+2]^{S} as

Λl,𝔫(z),f=1M(𝔫)δ(z=0)vS(𝔫)qv(z1)/21+qv(z+1)/2πΠcus(l,𝔫)W𝔫(z)(π)L(z+12,π;Ad)L(1,π;Ad)f(𝐱S(π)),fC(ΩS),formulae-sequencesuperscriptsubscriptΛ𝑙𝔫𝑧𝑓1Msuperscript𝔫𝛿𝑧0subscriptproduct𝑣𝑆𝔫superscriptsubscript𝑞𝑣𝑧121superscriptsubscript𝑞𝑣𝑧12subscript𝜋subscriptΠcus𝑙𝔫subscriptsuperscript𝑊𝑧𝔫𝜋𝐿𝑧12𝜋Ad𝐿1𝜋Ad𝑓subscript𝐱𝑆𝜋𝑓𝐶subscriptΩ𝑆\displaystyle\langle\Lambda_{l,{\mathfrak{n}}}^{(z)},f\rangle=\frac{1}{{\rm M}({\mathfrak{n}})^{\delta(z=0)}}\,\,\prod_{v\in S({\mathfrak{n}})}\frac{q_{v}^{(z-1)/2}}{1+q_{v}^{(z+1)/2}}\,\sum_{\pi\in\Pi_{\rm cus}(l,{\mathfrak{n}})}W^{(z)}_{{\mathfrak{n}}}(\pi)\frac{L(\tfrac{z+1}{2},\pi;{\rm Ad})}{L(1,\pi;{\rm Ad})}f({\bf x}_{S}(\pi)),\quad f\in C(\Omega_{S}),

where 𝐱S(π)={xv(π)}vSsubscript𝐱𝑆𝜋subscriptsubscript𝑥𝑣𝜋𝑣𝑆{\bf x}_{S}(\pi)=\{x_{v}(\pi)\}_{v\in S}, and M(𝔫)=vS(𝔫)logqv/(1+qv1/2)M𝔫subscript𝑣𝑆𝔫subscript𝑞𝑣1superscriptsubscript𝑞𝑣12{\rm M}({\mathfrak{n}})=\sum_{v\in S({\mathfrak{n}})}{\log q_{v}}/(1+q_{v}^{-1/2}). Note that 21logN(𝔫)M(𝔫)logN(𝔫)superscript21N𝔫M𝔫N𝔫2^{-1}\log{\operatorname{N}}({\mathfrak{n}})\leqslant{\rm M}({\mathfrak{n}})\leqslant\log{\operatorname{N}}({\mathfrak{n}}). The value L(z+12,π;Ad)𝐿𝑧12𝜋AdL(\tfrac{z+1}{2},\pi;{\operatorname{Ad}}) is non-negative for z𝑧z lying in Re(z)1Re𝑧1\operatorname{Re}(z)\geqslant 1, the closure of the absolute convergence region of the Euler product. In some cases, it is more enlightening to work with the assumption

(P) : L(s,π,Ad)0𝐿𝑠𝜋Ad0L(s,\pi,{\operatorname{Ad}})\geqslant 0 for any s[0,1)𝑠01s\in[0,1) and for all holomorphic cuspidal automorphic representations π𝜋\pi of PGL(2,𝔸)PGL2𝔸{\operatorname{PGL}}(2,{\mathbb{A}}) with weight l𝑙l,

which ensures that the measure Λl,𝔫(z)superscriptsubscriptΛ𝑙𝔫𝑧\Lambda_{l,{\mathfrak{n}}}^{(z)} is non-negative for z[0,1]𝑧01z\in[0,1]. The statement (P) is not proved up to now, however it is highly expected to be true from the entireness of L(s,π;Ad)𝐿𝑠𝜋AdL(s,\pi;{\operatorname{Ad}}) ([22], [6]) combined with the Riemann hypothesis of the L𝐿L-function L(s,π;Ad)𝐿𝑠𝜋AdL(s,\pi;{\operatorname{Ad}}).

For z[0,1]𝑧01z\in[0,1] and vS𝑣𝑆v\in S, we define a non-negative Radon measure on [2,2]22[-2,2] as

λv(z),fv=1+qv(z+1)/2π22fv(x)(1x2/4)1/2(qv(1+z)/4+qv(1+z)/4)2x2𝑑x,fvC([2,2]).formulae-sequencesuperscriptsubscript𝜆𝑣𝑧subscript𝑓𝑣1superscriptsubscript𝑞𝑣𝑧12𝜋superscriptsubscript22subscript𝑓𝑣𝑥superscript1superscript𝑥2412superscriptsuperscriptsubscript𝑞𝑣1𝑧4superscriptsubscript𝑞𝑣1𝑧42superscript𝑥2differential-d𝑥subscript𝑓𝑣𝐶22\displaystyle\langle\lambda_{v}^{(z)},f_{v}\rangle=\frac{1+q_{v}^{(z+1)/2}}{\pi}\int_{-2}^{2}f_{v}(x)\frac{(1-x^{2}/4)^{1/2}}{(q_{v}^{(1+z)/4}+q_{v}^{-(1+z)/4})^{2}-x^{2}}\,{{d}}x,\quad f_{v}\in C([-2,2]).

Note that the measure λv(1)superscriptsubscript𝜆𝑣1\lambda_{v}^{(1)} coincides with the limit measure in Serre’s theorem [19, Théorème 1]. For z[0,1]𝑧01z\in[0,1], set

Cl(z)=2DF3/2{21z2π3z+14Γ(z+34)}[F:]vΣΓ(lv+z12)4πΓ(lv1)superscriptsubscript𝐶𝑙𝑧2superscriptsubscript𝐷𝐹32superscriptsuperscript21𝑧2superscript𝜋3𝑧14Γ𝑧34delimited-[]:𝐹subscriptproduct𝑣subscriptΣΓsubscript𝑙𝑣𝑧124𝜋Γsubscript𝑙𝑣1\displaystyle C_{l}^{(z)}=2D_{F}^{3/2}\left\{2^{\frac{1-z}{2}}\pi^{-\frac{3z+1}{4}}\Gamma\left(\tfrac{z+3}{4}\right)\right\}^{[F:{\mathbb{Q}}]}\,\prod_{v\in\Sigma_{\infty}}\frac{\Gamma\left(l_{v}+\tfrac{z-1}{2}\right)}{4\pi\Gamma(l_{v}-1)}

and r(z)=ζF,fin(z+1)𝑟𝑧subscript𝜁𝐹fin𝑧1r(z)=\zeta_{F,{\rm fin}}(z+1) if z>0𝑧0z>0 and r(0)=Resz=1ζF,fin(z)𝑟0subscriptRes𝑧1subscript𝜁𝐹fin𝑧r(0)={\rm Res}_{z=1}\zeta_{F,{\rm fin}}(z). Let 𝒫(ΩS)𝒫subscriptΩ𝑆{\mathcal{P}}(\Omega_{S}) be the space of complex-valued polynomial functions on ΩSsubscriptΩ𝑆\Omega_{S}; by the Stone-Weierstrass theorem, 𝒫(ΩS)𝒫subscriptΩ𝑆{\mathcal{P}}(\Omega_{S}) is a dense subspace of C(𝔛S0)𝐶superscriptsubscript𝔛𝑆0C({\mathfrak{X}}_{S}^{0}). Fix a set of prime ideals 𝔞j(1jh)subscript𝔞𝑗1𝑗{\mathfrak{a}}_{j}\,(1\leqslant j\leqslant h) of 𝔬𝔬\mathfrak{o} different from 𝔭v(vS)subscript𝔭𝑣𝑣𝑆{\mathfrak{p}}_{v}\,(v\in S) and mapped bijectively to the ideal class group of F𝐹F; this is possible by Chebotarev’s density theorem.

Theorem 1.3.

Let F𝐹F be as in Corollary 1.2. Let l=(lv)vΣ(2)Σ𝑙subscriptsubscript𝑙𝑣𝑣subscriptΣsuperscript2subscriptΣl=(l_{v})_{v\in\Sigma_{\infty}}\in(2{\mathbb{N}})^{\Sigma_{\infty}} be such that l¯=minvΣlv4¯𝑙subscript𝑣subscriptΣsubscript𝑙𝑣4{\underline{l}}=\min_{v\in\Sigma_{\infty}}l_{v}\geqslant 4. Let S𝑆S be a finite set of non-dyadic finite places of F𝐹F. Let 𝔫𝔬𝔫𝔬{\mathfrak{n}}\subset\mathfrak{o} be a square-free ideal relatively prime to the ideals 2𝔬2𝔬2\mathfrak{o}, 𝔞i(1jh)subscript𝔞𝑖1𝑗{\mathfrak{a}}_{i}\,(1\leqslant j\leqslant h) and 𝔭v(vS)subscript𝔭𝑣𝑣𝑆{\mathfrak{p}}_{v}\,(v\in S).

  • (1)

    Let 0<σ<l¯30𝜎¯𝑙30<\sigma<\underline{l}-3. As N(𝔫)N𝔫{\operatorname{N}}({\mathfrak{n}})\rightarrow\infty,

    supz[0,min(1,σ)]|Λl,𝔫(z),fr(z)Cl(z)vSλv(z),f|0for any f𝒫(ΩS).\sup_{z\in[0,\min(1,\sigma)]}\left|\langle\Lambda_{l,{\mathfrak{n}}}^{(z)},f\rangle-r(z)C_{l}^{(z)}\langle\otimes_{v\in S}\lambda_{v}^{(z)},f\rangle\right|\rightarrow 0\quad\text{for any $f\in{\mathcal{P}}(\Omega_{S})$}.

    Under the condition (P), the same limit formulas are true for all fC(ΩS)𝑓𝐶subscriptΩ𝑆f\in C(\Omega_{S}), i.e., the measure Λl,𝔫(z)superscriptsubscriptΛ𝑙𝔫𝑧\Lambda_{l,{\mathfrak{n}}}^{(z)} converges *-weakly to r(z)Cl(z)vSλv(z)subscripttensor-product𝑣𝑆𝑟𝑧superscriptsubscript𝐶𝑙𝑧superscriptsubscript𝜆𝑣𝑧r(z)\,C_{l}^{(z)}\,\otimes_{v\in S}\lambda_{v}^{(z)} on 𝔛S0superscriptsubscript𝔛𝑆0{\mathfrak{X}}_{S}^{0}.

  • (2)

    Assume (P). Let Jv=[tv,tv](tv<tv,vS)subscript𝐽𝑣subscript𝑡𝑣superscriptsubscript𝑡𝑣formulae-sequencesubscript𝑡𝑣superscriptsubscript𝑡𝑣𝑣𝑆J_{v}=[t_{v},t_{v}^{\prime}]\,(t_{v}<t_{v}^{\prime},\,v\in S) be a family of closed subintervals of [2,2]22[-2,2] and 0<σ<l¯30𝜎¯𝑙30<\sigma<\underline{l}-3. There exists M>0𝑀0M>0 such that for any prime ideal 𝔫𝔫{\mathfrak{n}} with N(𝔫)>MN𝔫𝑀{\operatorname{N}}({\mathfrak{n}})>M and relatively prime to 2vS(𝔭v𝔬)j=1h𝔞j2subscriptproduct𝑣𝑆subscript𝔭𝑣𝔬superscriptsubscriptproduct𝑗1subscript𝔞𝑗2\prod_{v\in S}({\mathfrak{p}}_{v}\cap{\mathfrak{o}})\prod_{j=1}^{h}{\mathfrak{a}}_{j} and for any z[0,min(1,σ)]𝑧01𝜎z\in[0,\min(1,\sigma)], there exists πΠcus(l,𝔫)𝜋subscriptΠ𝑐𝑢𝑠𝑙𝔫\pi\in\Pi_{cus}(l,{\mathfrak{n}}) with the following properties: 𝔣π=𝔫subscript𝔣𝜋𝔫{\mathfrak{f}}_{\pi}={\mathfrak{n}}, L(z+12,π;Ad)0𝐿𝑧12𝜋Ad0L(\frac{z+1}{2},\pi;{\operatorname{Ad}})\not=0, and 𝐱S(π)Jsubscript𝐱𝑆𝜋𝐽{\mathbf{x}}_{S}(\pi)\in J. When J=ΩS𝐽subscriptΩ𝑆J=\Omega_{S}, the same conclusion holds without (P).

1.3. Organization of paper

In §2, we construct a kernel function 𝚽l(𝔫|𝐬;g,h)superscript𝚽𝑙conditional𝔫𝐬𝑔{\mathbf{\Phi}}^{l}({\mathfrak{n}}|{\mathbf{s}};g,h) on G𝔸×G𝔸subscript𝐺𝔸subscript𝐺𝔸G_{\mathbb{A}}\times G_{\mathbb{A}} which represents the resolvent vS{𝕋v(qv(1sv)/2+qv(1+sv)/2)}1subscriptproduct𝑣𝑆superscriptsubscript𝕋𝑣superscriptsubscript𝑞𝑣1subscript𝑠𝑣2superscriptsubscript𝑞𝑣1subscript𝑠𝑣21\prod_{v\in S}\{{\mathbb{T}}_{v}-(q_{v}^{(1-s_{v})/2}+q_{v}^{(1+s_{v})/2})\}^{-1} of the product of the shifted normalized Hecke operators at vS𝑣𝑆v\in S depending on a set of parameters 𝐬=(sv)vS𝐬subscriptsubscript𝑠𝑣𝑣𝑆{\mathbf{s}}=(s_{v})_{v\in S} and acting on the space of Hilbert modular cusp forms on G𝔸subscript𝐺𝔸G_{\mathbb{A}} of a fixed weight invariant by the open compact subgroup 𝕂0(𝔫)subscript𝕂0𝔫{\mathbb{K}}_{0}({\mathfrak{n}}). In §3, we introduce the smoothed Eisenstein series β(g)superscriptsubscript𝛽𝑔{\mathcal{E}}_{\beta}^{*}(g) on G𝔸subscript𝐺𝔸G_{\mathbb{A}} depending on an entire function β(z)𝛽𝑧\beta(z) vertically of rapid decay. Our definition is similar to but a bit different from the usual definition of the wave-packet in that we start from the normalized Eisenstein series. By a constraint on β(z)𝛽𝑧\beta(z) posed to cancel the poles of the Eisenstein series, our β(g)superscriptsubscript𝛽𝑔{\mathcal{E}}_{\beta}^{*}(g) is shown to be rapidly decreasing on a Siegel domain (Proposition 3.2). As in [12], we compute the inner-product of βsuperscriptsubscript𝛽{\mathcal{E}}_{\beta}^{*} and the diagonal restriction of 𝚽l(𝔫|𝐬;g,g)superscript𝚽𝑙conditional𝔫𝐬𝑔𝑔{\mathbf{\Phi}}^{l}({\mathfrak{n}}|{\mathbf{s}};g,g) on Z𝔸GF\G𝔸\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}} in two ways obtaining its two expressions referred to the spectral side and the geometric side. The spectral side is constructed in §3 without any difficulty by the Rankin-Selberg integral; the main issue here is the calculation of local zeta integrals for old forms. The computation of the geometric side is harder to accomplish. Since our test function is not compactly supported, the argument in [12] to deal with the unipotent and the hyperbolic terms by the Poisson summation formula is not applied as it is. By a counter shifting argument inspired by a similar analysis in [21], we entirely circumvent the usage of the Poisson summation formula. The computations of unipotent terms and hyperbolic terms are done in §5 and §6, respectively: We prove the absolute convergence of the hyperbolic term by estimating local orbital integrals whose exact formulas are calculated in §10. By the local multiplicity one theorem of the Waldspurger model ([32, Proposition 9’], [33], [34], [4, §1]), the computation of the elliptic term boils down to the determination of the Eisenstein periods along elliptic tori and the calculation of local orbital integrals. In §7, we recall the formula of Eisenstein period originally due to Hecke, providing a proof in a modern style, and establish the absolute convergence of the elliptic term, granting the formulas of local orbital integrals to be proved in §10. The main results are proved in §8. In §9, we state Theorem 9.1 which should be regarded as a version of Gon’s formula. Since its proof is almost identical to Corollary 1.2 and is much easier than that in some aspect, only a brief indication for proof will be given there. We suggest the readers to move to § 9 to see the statement of Theorem 9.1 after finishing this introduction. Theorem 9.1 has an application to non-vanishing L𝐿L-values, which we refer to our forthcoming work [28].

1.4. Notation

Throughout this paper, we adapt Vinogradov’s notation much-less-than\ll. For any complex-valued functions f𝑓f and g𝑔g on a set X𝑋X, we write f(x)g(x)much-less-than𝑓𝑥𝑔𝑥f(x)\ll g(x) if there exists a constant C>0𝐶0C>0 independent of xX𝑥𝑋x\in X such that |f(x)|C|g(x)|𝑓𝑥𝐶𝑔𝑥|f(x)|\leqslant C|g(x)| for all x𝑥x. We write f(x)g(x)asymptotically-equals𝑓𝑥𝑔𝑥f(x)\asymp g(x) when both f(x)g(x)much-less-than𝑓𝑥𝑔𝑥f(x)\ll g(x) and g(x)f(x)much-less-than𝑔𝑥𝑓𝑥g(x)\ll f(x) hold. If we emphasize dependence of the implied constant C𝐶C on some parameters a,b,c,𝑎𝑏𝑐a,b,c,\ldots, we write f(x)a,b,c,g(x)subscriptmuch-less-than𝑎𝑏𝑐𝑓𝑥𝑔𝑥f(x)\ll_{a,b,c,\ldots}g(x).

2. Construction of the kernel function

The Haar measures we work with in this article are fixed in the following way. For vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}, let dxv𝑑subscript𝑥𝑣{{d}}x_{v} be the additive Haar measure of Fvsubscript𝐹𝑣F_{v} such that 𝔬v𝑑xv=qvdv/2subscriptsubscript𝔬𝑣differential-dsubscript𝑥𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣2\int_{\mathfrak{o}_{v}}{{d}}x_{v}=q_{v}^{-d_{v}/2} if vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and |x|v<1𝑑xv=2subscriptsubscript𝑥𝑣1differential-dsubscript𝑥𝑣2\int_{|x|_{v}<1}{{d}}x_{v}=2 if vΣ𝑣subscriptΣv\in\Sigma_{\infty}, where dvsubscript𝑑𝑣d_{v} is the exponent of the local different of Fvsubscript𝐹𝑣F_{v}. Fix measures d×xvsuperscript𝑑subscript𝑥𝑣{{d}}^{\times}x_{v} on Fv×subscriptsuperscript𝐹𝑣F^{\times}_{v} and d×xsuperscript𝑑𝑥{{d}}^{\times}x on 𝔸×superscript𝔸{\mathbb{A}}^{\times} by d×xv=ζFv(1)dxv/|xv|vsuperscript𝑑subscript𝑥𝑣subscript𝜁subscript𝐹𝑣1𝑑subscript𝑥𝑣subscriptsubscript𝑥𝑣𝑣{{d}}^{\times}x_{v}=\zeta_{F_{v}}(1)\,{{d}}x_{v}/|x_{v}|_{v} and by d×x=vΣFd×xvsuperscript𝑑𝑥subscriptproduct𝑣subscriptΣ𝐹superscript𝑑subscript𝑥𝑣{{d}}^{\times}x=\prod_{v\in\Sigma_{F}}{{d}}^{\times}x_{v}, respectively. We fix Haar measures on ZvFv×subscript𝑍𝑣superscriptsubscript𝐹𝑣Z_{v}\cong F_{v}^{\times} and Z𝔸𝔸×subscript𝑍𝔸superscript𝔸Z_{\mathbb{A}}\cong{\mathbb{A}}^{\times} accordingly. We endow Hvsubscript𝐻𝑣H_{v}, Nvsubscript𝑁𝑣N_{v}, and 𝕂vsubscript𝕂𝑣{\mathbb{K}}_{v} with measures dhv𝑑subscript𝑣{{d}}h_{v}, dnv𝑑subscript𝑛𝑣{{d}}n_{v}, and dkv𝑑subscript𝑘𝑣{{d}}k_{v}, respectively by setting dhv=d×t1,vd×t2,v𝑑subscript𝑣superscript𝑑subscript𝑡1𝑣superscript𝑑subscript𝑡2𝑣{{d}}h_{v}={{d}}^{\times}t_{1,v}\,{{d}}^{\times}t_{2,v} if hv=[t1,v00t2,v]subscript𝑣delimited-[]subscript𝑡1𝑣00subscript𝑡2𝑣h_{v}=\left[\begin{smallmatrix}t_{1,v}&0\\ 0&t_{2,v}\end{smallmatrix}\right], dnv=dxv𝑑subscript𝑛𝑣𝑑subscript𝑥𝑣{{d}}n_{v}={{d}}x_{v} if nv=[1xv01]subscript𝑛𝑣delimited-[]1subscript𝑥𝑣01n_{v}=\left[\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right], and by requiring vol(𝕂v,dkv)=1volsubscript𝕂𝑣𝑑subscript𝑘𝑣1{\operatorname{vol}}({\mathbb{K}}_{v},dk_{v})=1. Then our Haar measure on Gvsubscript𝐺𝑣G_{v} is defined as dgv=dhvdnvdkv𝑑subscript𝑔𝑣𝑑subscript𝑣𝑑subscript𝑛𝑣𝑑subscript𝑘𝑣{{d}}g_{v}={{d}}h_{v}\,{{d}}n_{v}\,{{d}}k_{v} by the Iwasawa decomposition Gv=HvNv𝕂vsubscript𝐺𝑣subscript𝐻𝑣subscript𝑁𝑣subscript𝕂𝑣G_{v}=H_{v}N_{v}{\mathbb{K}}_{v}. Note that vol(𝐊v,dgv)=qv3dv/2volsubscript𝐊𝑣𝑑subscript𝑔𝑣superscriptsubscript𝑞𝑣3subscript𝑑𝑣2{\operatorname{vol}}({\mathbf{K}}_{v},dg_{v})=q_{v}^{-3d_{v}/2}. On 𝕂𝕂{\mathbb{K}}, N𝔸subscript𝑁𝔸N_{\mathbb{A}}, H𝔸subscript𝐻𝔸H_{\mathbb{A}}, Z𝔸subscript𝑍𝔸Z_{\mathbb{A}} and G𝔸subscript𝐺𝔸G_{\mathbb{A}}, we always use the product measures of their factors. Let chXsubscriptch𝑋{\rm ch}_{X} denote the characteristic function of a set X𝑋X. We omit its domain as we guess easily from context.

2.1. Convergence lemmas

Let Ad:GGL(𝔰𝔩2):Ad𝐺GL𝔰subscript𝔩2{\operatorname{Ad}}:G\rightarrow{\operatorname{GL}}({\mathfrak{s}}{\mathfrak{l}}_{2}) be the adjoint representation of G𝐺G on 𝔰𝔩2𝔰subscript𝔩2{\mathfrak{sl}}_{2}. For gG𝑔𝐺g\in G, let (Ad(g)ij)1i,j3({\operatorname{Ad}}(g)_{ij})_{1\leqslant i,j\leqslant 3} be the representing matrix with respect to the F𝐹F-basis {[1001],[0100],[0010]}delimited-[]1001delimited-[]0100delimited-[]0010\{\left[\begin{smallmatrix}1&0\\ 0&-1\end{smallmatrix}\right],\,\left[\begin{smallmatrix}0&1\\ 0&0\end{smallmatrix}\right],\,\left[\begin{smallmatrix}0&0\\ 1&0\end{smallmatrix}\right]\} of 𝔰𝔩2(F)𝔰subscript𝔩2𝐹{\mathfrak{sl}}_{2}(F). For gvGvsubscript𝑔𝑣subscript𝐺𝑣g_{v}\in G_{v}, set

gvv={max{|Ad(gv)ij|v| 1i,j3},(vΣfin),{ij|Ad(gv)ij|v2}1/2,(vΣ).\displaystyle\|g_{v}\|_{v}=\begin{cases}&\max\{|{\operatorname{Ad}}(g_{v})_{ij}|_{v}\,|\,1\leqslant i,j\leqslant 3\,\},\quad(v\in\Sigma_{\rm fin}),\\ &\{\sum_{ij}|{\operatorname{Ad}}(g_{v})_{ij}|_{v}^{2}\}^{1/2},\quad(v\in\Sigma_{\infty}).\end{cases}

Then, zvkvgvkvv=gvvsubscriptnormsubscript𝑧𝑣subscript𝑘𝑣subscript𝑔𝑣subscriptsuperscript𝑘𝑣𝑣subscriptnormsubscript𝑔𝑣𝑣\|z_{v}k_{v}g_{v}k^{\prime}_{v}\|_{v}=\|g_{v}\|_{v} for any zvZvsubscript𝑧𝑣subscript𝑍𝑣z_{v}\in Z_{v} and kv,kv𝕂vsubscript𝑘𝑣superscriptsubscript𝑘𝑣subscript𝕂𝑣k_{v},k_{v}^{\prime}\in{\mathbb{K}}_{v}. The norm of an adele point g=(gv)G𝔸𝑔subscript𝑔𝑣subscript𝐺𝔸g=(g_{v})\in G_{\mathbb{A}} is defined by g𝔸=vΣFgvvsubscriptnorm𝑔𝔸subscriptproduct𝑣subscriptΣ𝐹subscriptnormsubscript𝑔𝑣𝑣\|g\|_{\mathbb{A}}=\prod_{v\in\Sigma_{F}}\|g_{v}\|_{v}. We remark that the norm g𝔸subscriptnorm𝑔𝔸\|g\|_{\mathbb{A}} is a Z𝔸subscript𝑍𝔸Z_{\mathbb{A}}-invariant and bi-𝕂𝕂{\mathbb{K}}-invariant function on G𝔸subscript𝐺𝔸G_{\mathbb{A}} taking values in [1,+)1[1,+\infty). It satisfies gh𝔸g𝔸h𝔸subscriptnorm𝑔𝔸subscriptnorm𝑔𝔸subscriptnorm𝔸\|gh\|_{\mathbb{A}}\leqslant\|g\|_{\mathbb{A}}\|h\|_{\mathbb{A}} for all g,hG𝔸𝑔subscript𝐺𝔸g,\,h\in G_{\mathbb{A}} and g𝔸y(g)asymptotically-equalssubscriptnorm𝑔𝔸𝑦𝑔\|g\|_{\mathbb{A}}\asymp y(g), g𝔖1𝑔superscript𝔖1g\in{\mathfrak{S}}^{1}, where 𝔖1superscript𝔖1{\mathfrak{S}}^{1} is a Siegel domain of G𝔸1={gG𝔸||detg|𝔸=1}superscriptsubscript𝐺𝔸1conditional-set𝑔subscript𝐺𝔸subscript𝑔𝔸1G_{\mathbb{A}}^{1}=\{g\in G_{\mathbb{A}}|\,|\det g|_{\mathbb{A}}=1\,\} and y:G𝔸+:𝑦subscript𝐺𝔸subscripty:G_{\mathbb{A}}\rightarrow{\mathbb{R}}_{+} is defined as

y([ab0d]k)=|a/d|𝔸,[ab0d]B𝔸,k𝕂.formulae-sequence𝑦delimited-[]𝑎𝑏0𝑑𝑘subscript𝑎𝑑𝔸formulae-sequencedelimited-[]𝑎𝑏0𝑑subscript𝐵𝔸𝑘𝕂y\left(\left[\begin{smallmatrix}a&b\\ 0&d\end{smallmatrix}\right]k\right)=|a/d|_{\mathbb{A}},\quad\left[\begin{smallmatrix}a&b\\ 0&d\end{smallmatrix}\right]\in B_{\mathbb{A}},\,k\in{\mathbb{K}}.
Lemma 2.1.

For σ𝜎\sigma\in{\mathbb{R}} and hG𝔸subscript𝐺𝔸h\in G_{\mathbb{A}}, set Ξσ(h)=γZF\GFγh𝔸σsubscriptΞ𝜎subscript𝛾\subscript𝑍𝐹subscript𝐺𝐹superscriptsubscriptnorm𝛾𝔸𝜎\Xi_{\sigma}(h)=\sum_{\gamma\in Z_{F}\backslash G_{F}}\|\gamma h\|_{\mathbb{A}}^{-\sigma}. If σ>1𝜎1\sigma>1, then the series Ξσ(h)subscriptΞ𝜎\Xi_{\sigma}(h) converges absolutely and locally uniformly in hG𝔸subscript𝐺𝔸h\in G_{\mathbb{A}}; moreover, the function hΞσ(h)maps-tosubscriptΞ𝜎h\mapsto\Xi_{\sigma}(h) is bounded on G𝔸subscript𝐺𝔸G_{\mathbb{A}}.

Proof.

Set G¯=Z\G¯𝐺\𝑍𝐺\overline{G}=Z\backslash G. Let 𝒰𝒰{\mathcal{U}} be a compact neighborhood of the identity of G¯𝔸=Z𝔸\G𝔸subscript¯𝐺𝔸\subscript𝑍𝔸subscript𝐺𝔸{\overline{G}}_{\mathbb{A}}=Z_{\mathbb{A}}\backslash G_{\mathbb{A}} such that G¯F𝒰1𝒰={12}subscript¯𝐺𝐹superscript𝒰1𝒰subscript12{\overline{G}}_{F}\cap{\mathcal{U}}^{-1}{\mathcal{U}}=\{1_{2}\}. Then γG¯F𝒰γhsubscript𝛾subscript¯𝐺𝐹𝒰𝛾\bigcup_{\gamma\in{\overline{G}}_{F}}{\mathcal{U}}\gamma h is a disjoint union for any hG𝔸subscript𝐺𝔸h\in G_{\mathbb{A}}. Set c0=supg𝒰g𝔸subscript𝑐0subscriptsupremum𝑔𝒰subscriptnorm𝑔𝔸c_{0}=\sup_{g\in{\mathcal{U}}}\|g\|_{\mathbb{A}}. We have gγh𝔸g𝔸γh𝔸c0γh𝔸subscriptnorm𝑔𝛾𝔸subscriptnorm𝑔𝔸subscriptnorm𝛾𝔸subscript𝑐0subscriptnorm𝛾𝔸\|g\gamma h\|_{\mathbb{A}}\leqslant\|g\|_{\mathbb{A}}\|\gamma h\|_{\mathbb{A}}\leqslant c_{0}\|\gamma h\|_{\mathbb{A}}, and thus

γh𝔸σc0σgγh𝔸σ(σ>0)superscriptsubscriptnorm𝛾𝔸𝜎superscriptsubscript𝑐0𝜎superscriptsubscriptnorm𝑔𝛾𝔸𝜎𝜎0\|\gamma h\|_{\mathbb{A}}^{-\sigma}\leqslant c_{0}^{\sigma}\|g\gamma h\|_{\mathbb{A}}^{-\sigma}\quad(\sigma>0)

for any g𝒰,hG¯𝔸formulae-sequence𝑔𝒰subscript¯𝐺𝔸g\in{\mathcal{U}},\,h\in{\overline{G}}_{\mathbb{A}} and γG¯F𝛾subscript¯𝐺𝐹\gamma\in{\overline{G}}_{F}. From this,

c0σvol(𝒰)Ξσ(h)superscriptsubscript𝑐0𝜎vol𝒰subscriptΞ𝜎absent\displaystyle c_{0}^{-\sigma}{\operatorname{vol}}({\mathcal{U}})\,\Xi_{\sigma}(h)\leqslant γG¯F𝒰gγh𝔸σ𝑑g=γG¯F𝒰γhg𝔸σ𝑑gG¯𝔸g𝔸σ𝑑g.subscript𝛾subscript¯𝐺𝐹subscript𝒰superscriptsubscriptnorm𝑔𝛾𝔸𝜎differential-d𝑔subscriptsubscript𝛾subscript¯𝐺𝐹𝒰𝛾superscriptsubscriptnorm𝑔𝔸𝜎differential-d𝑔subscriptsubscript¯𝐺𝔸superscriptsubscriptnorm𝑔𝔸𝜎differential-d𝑔\displaystyle\sum_{\gamma\in{\overline{G}}_{F}}\textstyle{\int}_{{\mathcal{U}}}\|g\gamma h\|_{\mathbb{A}}^{-\sigma}\,{{d}}g=\textstyle{\int}_{\bigcup_{\gamma\in{\overline{G}}_{F}}{\mathcal{U}}\gamma h}\|g\|_{\mathbb{A}}^{-\sigma}\,{{d}}g\leqslant\textstyle{\int}_{{\overline{G}}_{\mathbb{A}}}\|g\|_{\mathbb{A}}^{-\sigma}\,{{d}}g.

Thus it suffices to show that the function gg𝔸σmaps-to𝑔superscriptsubscriptnorm𝑔𝔸𝜎g\mapsto\|g\|_{\mathbb{A}}^{-\sigma} on Z𝔸\G𝔸\subscript𝑍𝔸subscript𝐺𝔸Z_{\mathbb{A}}\backslash G_{\mathbb{A}} is integrable if σ>1𝜎1\sigma>1. For σ𝜎\sigma\in{\mathbb{R}}, set 𝔍v(σ)=Zv\Gvgvvσ𝑑gv.subscript𝔍𝑣𝜎subscript\subscript𝑍𝑣subscript𝐺𝑣superscriptsubscriptnormsubscript𝑔𝑣𝑣𝜎differential-dsubscript𝑔𝑣{\mathfrak{J}}_{v}(\sigma)=\int_{Z_{v}\backslash G_{v}}\|g_{v}\|_{v}^{-\sigma}\,{{d}}g_{v}. If vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and σ>0𝜎0\sigma>0, then by the Cartan decomposition Zv\Gv=n0𝕂v[ϖvn001]𝕂v\subscript𝑍𝑣subscript𝐺𝑣subscript𝑛subscript0subscript𝕂𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣𝑛001subscript𝕂𝑣Z_{v}\backslash G_{v}=\bigcup_{n\in{\mathbb{N}}_{0}}{\mathbb{K}}_{v}\left[\begin{smallmatrix}\varpi_{v}^{n}&0\\ 0&1\end{smallmatrix}\right]{\mathbb{K}}_{v}, we have

𝔍v(σ)subscript𝔍𝑣𝜎\displaystyle{\mathfrak{J}}_{v}(\sigma) =n=0qvnσvol(𝕂v[ϖvn001]𝕂v)=1+n=1qvnσqvn(1+qv1)=(1+qvσ1)/(1qvσ).absentsuperscriptsubscript𝑛0superscriptsubscript𝑞𝑣𝑛𝜎volsubscript𝕂𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣𝑛001subscript𝕂𝑣1superscriptsubscript𝑛1superscriptsubscript𝑞𝑣𝑛𝜎superscriptsubscript𝑞𝑣𝑛1superscriptsubscript𝑞𝑣11superscriptsubscript𝑞𝑣𝜎11superscriptsubscript𝑞𝑣𝜎\displaystyle=\sum_{n=0}^{\infty}q_{v}^{-n\sigma}{\operatorname{vol}}\left({\mathbb{K}}_{v}\left[\begin{smallmatrix}\varpi_{v}^{n}&0\\ 0&1\end{smallmatrix}\right]{\mathbb{K}}_{v}\right)=1+\sum_{n=1}^{\infty}q_{v}^{-n\sigma}\,q_{v}^{n}(1+q_{v}^{-1})=(1+q_{v}^{-\sigma-1})/(1-q_{v}^{-\sigma}).

From this, the convergence of vΣfin𝔍v(σ)subscriptproduct𝑣subscriptΣfinsubscript𝔍𝑣𝜎\prod_{v\in\Sigma_{{\rm fin}}}{\mathfrak{J}}_{v}(\sigma) for σ>1𝜎1\sigma>1 is evident. If vΣ𝑣subscriptΣv\in\Sigma_{\infty} and σ>1𝜎1\sigma>1, then by the Cartan decomposition Gv=𝕂vHv𝕂vsubscript𝐺𝑣subscript𝕂𝑣subscript𝐻𝑣subscript𝕂𝑣G_{v}={\mathbb{K}}_{v}H_{v}{\mathbb{K}}_{v}, we have

𝔍v(σ)subscript𝔍𝑣𝜎\displaystyle{\mathfrak{J}}_{v}(\sigma) =4π0[et00et]σsinh 2tdt=4π0(e4t+e4t+1)σ/2sinh 2tdt<+.absent4𝜋superscriptsubscript0superscriptnormdelimited-[]superscript𝑒𝑡00superscript𝑒𝑡𝜎2𝑡𝑑𝑡4𝜋superscriptsubscript0superscriptsuperscript𝑒4𝑡superscript𝑒4𝑡1𝜎22𝑡𝑑𝑡\displaystyle=4\pi\textstyle{\int}_{0}^{\infty}\left\|\left[\begin{smallmatrix}e^{t}&0\\ 0&e^{-t}\end{smallmatrix}\right]\right\|^{-\sigma}\,{\sinh\,}2t\,{{d}}t=4\pi\textstyle{\int}_{0}^{\infty}(e^{4t}+e^{-4t}+1)^{-\sigma/2}\,{\sinh\,}2t\,{{d}}t<+\infty.

This completes the proof of the pointwise convergence. It remains to confirm the local uniformity of the convergence. Let U𝑈U be a compact subset of G¯𝔸subscript¯𝐺𝔸{\overline{G}}_{\mathbb{A}}. Then, γ𝔸(suphUh1𝔸)γh𝔸subscriptnorm𝛾𝔸subscriptsupremum𝑈subscriptnormsuperscript1𝔸subscriptnorm𝛾𝔸\|\gamma\|_{\mathbb{A}}\leqslant\,(\sup_{h\in U}\|h^{-1}\|_{\mathbb{A}})\|\gamma h\|_{\mathbb{A}} for any γG¯F𝛾subscript¯𝐺𝐹\gamma\in{\overline{G}}_{F} and hU𝑈h\in U. From this, Ξσ(h)subscriptΞ𝜎\Xi_{\sigma}(h) (hU)𝑈(h\in U) is majorized term-wisely by the convergent series Ξσ(12)subscriptΞ𝜎subscript12\Xi_{\sigma}(1_{2}) for σ>1𝜎1\sigma>1. ∎

Proposition 2.2.

Let φ:G𝔸:𝜑subscript𝐺𝔸\varphi:G_{\mathbb{A}}\rightarrow{\mathbb{C}} be a function such that φ(zg)=φ(g)𝜑𝑧𝑔𝜑𝑔\varphi(zg)=\varphi(g) for all zZ𝔸𝑧subscript𝑍𝔸z\in Z_{\mathbb{A}} and |φ(g)|g𝔸mmuch-less-than𝜑𝑔superscriptsubscriptnorm𝑔𝔸𝑚|\varphi(g)|\ll\|g\|_{\mathbb{A}}^{-m} on G𝔸subscript𝐺𝔸G_{\mathbb{A}} with m>1𝑚1m>1. Then, the series

Kφ(g,h)=γZF\GFφ(g1γh),(g,h)G𝔸×G𝔸formulae-sequencesubscript𝐾𝜑𝑔subscript𝛾\subscript𝑍𝐹subscript𝐺𝐹𝜑superscript𝑔1𝛾𝑔subscript𝐺𝔸subscript𝐺𝔸K_{\varphi}(g,h)=\sum_{\gamma\in Z_{F}\backslash G_{F}}\varphi(g^{-1}\gamma h),\quad(g,h)\in G_{\mathbb{A}}\times G_{\mathbb{A}}

converges absolutely and locally uniformly. Moreover, the following holds.

  • (i)

    suphG𝔸|Kφ(g,h)|g𝔸m,gG𝔸formulae-sequencemuch-less-thansubscriptsupremumsubscript𝐺𝔸subscript𝐾𝜑𝑔superscriptsubscriptnorm𝑔𝔸𝑚𝑔subscript𝐺𝔸\sup_{h\in G_{\mathbb{A}}}|K_{\varphi}(g,h)|\ll\|g\|_{\mathbb{A}}^{m},\quad g\in G_{\mathbb{A}} with the implied constant independent of g𝑔g.

  • (ii)

    For any gG𝔸𝑔subscript𝐺𝔸g\in G_{\mathbb{A}}, the function Kφ(g,)subscript𝐾𝜑𝑔K_{\varphi}(g,-) belongs to Lq(Z𝔸GF\G𝔸)superscript𝐿𝑞\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸L^{q}(Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}) for any q>0𝑞0q>0.

Proof.

Let U𝑈U be a compact subset of G¯𝔸subscript¯𝐺𝔸{\overline{G}}_{\mathbb{A}}. Then

|φ(g1γh)|g1γh𝔸mγ𝔸m,(g,h)U×U,γG¯F.formulae-sequencemuch-less-than𝜑superscript𝑔1𝛾superscriptsubscriptnormsuperscript𝑔1𝛾𝔸𝑚asymptotically-equalssuperscriptsubscriptnorm𝛾𝔸𝑚formulae-sequence𝑔𝑈𝑈𝛾subscript¯𝐺𝐹|\varphi(g^{-1}\gamma h)|\ll\|g^{-1}\gamma h\|_{\mathbb{A}}^{-m}\asymp\|\gamma\|_{\mathbb{A}}^{-m},\quad(g,h)\in U\times U,\,\gamma\in{\overline{G}}_{F}.

From this, the series γG¯F|φ(g1γh)|subscript𝛾subscript¯𝐺𝐹𝜑superscript𝑔1𝛾\sum_{\gamma\in\overline{G}_{F}}|\varphi(g^{-1}\gamma h)| (g,hU)𝑔𝑈(g,h\in U) is dominated by Ξm(12)subscriptΞ𝑚subscript12\Xi_{m}(1_{2}), which is convergent if m>1𝑚1m>1 by Lemma 2.1.

(i) From γh𝔸g𝔸g1γh𝔸subscriptnorm𝛾𝔸subscriptnorm𝑔𝔸subscriptnormsuperscript𝑔1𝛾𝔸\|\gamma h\|_{\mathbb{A}}\leqslant\|g\|_{\mathbb{A}}\|g^{-1}\gamma h\|_{\mathbb{A}}, we have

γG¯F|φ(g1γh)|γG¯Fg1γh𝔸mg𝔸mΞm(h)much-less-thansubscript𝛾subscript¯𝐺𝐹𝜑superscript𝑔1𝛾subscript𝛾subscript¯𝐺𝐹superscriptsubscriptnormsuperscript𝑔1𝛾𝔸𝑚superscriptsubscriptnorm𝑔𝔸𝑚subscriptΞ𝑚\displaystyle\sum_{\gamma\in{\overline{G}}_{F}}|\varphi(g^{-1}\gamma h)|\ll\sum_{\gamma\in{\overline{G}}_{F}}\|g^{-1}\gamma h\|_{\mathbb{A}}^{-m}\leqslant\|g\|_{\mathbb{A}}^{m}\,\Xi_{m}(h)

for any g,hG¯𝔸𝑔subscript¯𝐺𝔸g,\,h\in{\overline{G}}_{\mathbb{A}}. Since Ξm(h)subscriptΞ𝑚\Xi_{m}(h) is bounded in hG¯𝔸subscript¯𝐺𝔸h\in{\overline{G}}_{\mathbb{A}}, we are done.

(ii) From (i), for a fixed g𝑔g, the function Kφ(g,h)subscript𝐾𝜑𝑔K_{\varphi}(g,h) in hh is bounded. Since Z𝔸GF\G𝔸\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}} is of finite volume, Kφ(g,)Lq(Z𝔸GF\G𝔸)subscript𝐾𝜑𝑔superscript𝐿𝑞\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸K_{\varphi}(g,-)\in L^{q}(Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}) for any q>0𝑞0q>0. ∎

2.2. Matrix coefficients of the discrete series

We recall a basic material from [14, §11, 14] to fix notation. For l2𝑙2l\in 2{\mathbb{N}}, let (δl,Dl)subscript𝛿𝑙subscript𝐷𝑙(\delta_{l},D_{l}) be the discrete series representation of PGL(2,)PGL2{\operatorname{PGL}}(2,{\mathbb{R}}) with minimal SO(2)SO2{\operatorname{SO}}(2)-type τlsubscript𝜏𝑙\tau_{l}, where τlsubscript𝜏𝑙\tau_{l} is the character of SO(2)SO2{\operatorname{SO}}(2) given by τl(cosθsinθsinθcosθ)=eilθsubscript𝜏𝑙𝜃𝜃𝜃𝜃superscript𝑒𝑖𝑙𝜃\tau_{l}(\begin{smallmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{smallmatrix})=e^{il\theta}. The matrix coefficient of Dlsubscript𝐷𝑙D_{l} is defined by

Φl(g)=(δl(g)vl|vl),gPGL(2,),formulae-sequencesuperscriptΦ𝑙𝑔conditionalsubscript𝛿𝑙𝑔subscript𝑣𝑙subscript𝑣𝑙𝑔PGL2\Phi^{l}(g)=(\delta_{l}(g)v_{l}|v_{l}),\qquad g\in{\operatorname{PGL}}(2,{\mathbb{R}}),

where vlsubscript𝑣𝑙v_{l} is a unit vector belonging to τlsubscript𝜏𝑙\tau_{l}. It satisfies the conditions :

  • (a)

    R(W)Φl(g)=0𝑅𝑊superscriptΦ𝑙𝑔0R({W})\Phi^{l}(g)=0 with W=12(1ii1)𝑊121𝑖𝑖1W=\frac{1}{2}\left(\begin{smallmatrix}1&-i\\ -i&-1\end{smallmatrix}\right),

  • (b)

    Φl(kgk)=τl(k)τl(k)Φl(g)superscriptΦ𝑙𝑘𝑔superscript𝑘subscript𝜏𝑙superscript𝑘subscript𝜏𝑙𝑘superscriptΦ𝑙𝑔\Phi^{l}(kgk^{\prime})=\tau_{l}(k^{\prime})\,\tau_{l}(k)\,\Phi^{l}(g) for k,kSO(2)𝑘superscript𝑘SO2k,k^{\prime}\in{\operatorname{SO}}(2),

  • (c)

    Φl(12)=1superscriptΦ𝑙subscript121\Phi^{l}(1_{2})=1.

These conditions uniquely determine the Csuperscript𝐶C^{\infty}-function ΦlsuperscriptΦ𝑙\Phi^{l} as

(2.1) Φl(g)=δ(detg>0)(4detg)l/2{(a+d)i(bc)}lforg=[abcd].formulae-sequencesuperscriptΦ𝑙𝑔𝛿𝑔0superscript4𝑔𝑙2superscript𝑎𝑑𝑖𝑏𝑐𝑙for𝑔delimited-[]𝑎𝑏𝑐𝑑\displaystyle\Phi^{l}(g)=\delta(\det g>0)\,(4\det g)^{l/2}\{(a+d)-i(b-c)\}^{-l}\quad\text{for}\,g=\left[\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right].

Let vΣ𝑣subscriptΣv\in\Sigma_{\infty}. We denote by ΦvlsuperscriptsubscriptΦ𝑣𝑙\Phi_{v}^{l} the function ΦlsuperscriptΦ𝑙\Phi^{l} on GvGL(2,)subscript𝐺𝑣GL2G_{v}\cong{\operatorname{GL}}(2,{\mathbb{R}}). Then,

(2.2) |Φvl(g)|gvvl/2,gGv.formulae-sequencemuch-less-thansuperscriptsubscriptΦ𝑣𝑙𝑔superscriptsubscriptnormsubscript𝑔𝑣𝑣𝑙2𝑔subscript𝐺𝑣\displaystyle|\Phi_{v}^{l}(g)|\ll\|g_{v}\|_{v}^{-l/2},\quad g\in G_{v}.
Lemma 2.3.

Let l2𝑙2l\in 2{\mathbb{N}}. Let vΣ𝑣subscriptΣv\in\Sigma_{\infty} and φ:Gv:𝜑subscript𝐺𝑣\varphi:G_{v}\rightarrow{\mathbb{C}} a Csuperscript𝐶C^{\infty}-function such that φ(zgk)=τl(k)φ(g)𝜑𝑧𝑔𝑘subscript𝜏𝑙𝑘𝜑𝑔\varphi(zgk)=\tau_{l}(k)\varphi(g) for (z,k)Zv×𝕂v0𝑧𝑘subscript𝑍𝑣superscriptsubscript𝕂𝑣0(z,k)\in Z_{v}\times{\mathbb{K}}_{v}^{0} and such that R(Wv)φ=0𝑅subscript𝑊𝑣𝜑0R(W_{v})\varphi=0. Then we have the relation

Zv\GvΦvl(hv)φ(hv)¯𝑑hv=4πl1φ(12)¯.subscript\subscript𝑍𝑣subscript𝐺𝑣superscriptsubscriptΦ𝑣𝑙subscript𝑣¯𝜑subscript𝑣differential-dsubscript𝑣4𝜋𝑙1¯𝜑subscript12\int_{Z_{v}\backslash G_{v}}\Phi_{v}^{l}(h_{v})\,\overline{\varphi(h_{v})}\,{{d}}h_{v}=\tfrac{4\pi}{l-1}\,\overline{\varphi(1_{2})}.
Proof.

Set fφ(hv)=vol(𝐊v0)1𝕂v0φ(khv)τl(k)dkf_{\varphi}(h_{v})={\operatorname{vol}}({\mathbf{K}}_{v}^{0})^{-1}\int_{{\mathbb{K}}_{v}^{0}}\varphi(kh_{v})\,\tau_{-l}(k)\,{{d}}k for hvGvsubscript𝑣subscript𝐺𝑣h_{v}\in G_{v}, where 𝕂v0SO(2)superscriptsubscript𝕂𝑣0SO2{\mathbb{K}}_{v}^{0}\cong{\operatorname{SO}}(2). Then fφsubscript𝑓𝜑f_{\varphi} satisfies the same conditions (a) and (b) as ΦlsuperscriptΦ𝑙\Phi^{l}. By the uniqueness of the solution to the differential equation (a), there is a constant C𝐶C such that fφ(hv)=CΦvl(hv)subscript𝑓𝜑subscript𝑣𝐶superscriptsubscriptΦ𝑣𝑙subscript𝑣f_{\varphi}(h_{v})=C\,{{\Phi_{v}^{l}(h_{v})}} for all hvGvsubscript𝑣subscript𝐺𝑣h_{v}\in G_{v}. By putting hv=12subscript𝑣subscript12h_{v}=1_{2}, we have C=fφ(12)=φ(12)𝐶subscript𝑓𝜑subscript12𝜑subscript12C=f_{\varphi}(1_{2})=\varphi(1_{2}). Substituting the equation fφ(h)¯=φ(12)¯Φvl(h)¯¯subscript𝑓𝜑¯𝜑subscript12¯superscriptsubscriptΦ𝑣𝑙\overline{f_{\varphi}(h)}=\overline{\varphi(1_{2})}\,\overline{\Phi_{v}^{l}(h)}, we have

Zv\GvΦvl(hv)φ(hv)¯𝑑hvsubscript\subscript𝑍𝑣subscript𝐺𝑣superscriptsubscriptΦ𝑣𝑙subscript𝑣¯𝜑subscript𝑣differential-dsubscript𝑣\displaystyle\textstyle{\int}_{Z_{v}\backslash G_{v}}\Phi_{v}^{l}(h_{v})\,\overline{\varphi(h_{v})}\,{{d}}h_{v} =Zv\GvΦvl(hv)fφ(hv)¯𝑑hv=φ(12)¯Zv\Gv|Φvl(hv)|2𝑑hv=4πl1φ(12)¯absentsubscript\subscript𝑍𝑣subscript𝐺𝑣superscriptsubscriptΦ𝑣𝑙subscript𝑣¯subscript𝑓𝜑subscript𝑣differential-dsubscript𝑣¯𝜑subscript12subscript\subscript𝑍𝑣subscript𝐺𝑣superscriptsuperscriptsubscriptΦ𝑣𝑙subscript𝑣2differential-dsubscript𝑣4𝜋𝑙1¯𝜑subscript12\displaystyle=\textstyle{\int}_{Z_{v}\backslash G_{v}}\Phi_{v}^{l}(h_{v})\,\overline{f_{\varphi}(h_{v})}\,{{d}}h_{v}=\overline{\varphi(1_{2})}\,\int_{Z_{v}\backslash G_{v}}|\Phi_{v}^{l}(h_{v})|^{2}\,{{d}}h_{v}=\tfrac{4\pi}{l-1}\,\overline{\varphi(1_{2})}

as desired. We refer to [14, Proposition 14.4] for the last identity. ∎

2.3. Green functions on GL(2)GL2{\operatorname{GL}}(2) over non-Archimedean local fields

Let vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. For s𝑠s\in{\mathbb{C}} such that Re(s)>1Re𝑠1\operatorname{Re}(s)>1, it is easy to show that there exists a unique function φ:Gv:𝜑subscript𝐺𝑣\varphi:G_{v}\rightarrow{\mathbb{C}} with the properties:

  • (a)

    φ(zkgk)=φ(g)𝜑𝑧𝑘𝑔superscript𝑘𝜑𝑔\varphi(zkgk^{\prime})=\varphi(g) for any k,k𝕂v𝑘superscript𝑘subscript𝕂𝑣k,k^{\prime}\in{\mathbb{K}}_{v}, zZv𝑧subscript𝑍𝑣z\in Z_{v}.

  • (b)

    [R(𝕋v(qv(1+s)/2+qv(1s)/2) 1𝕂v)]φ=chZv𝕂vdelimited-[]𝑅subscript𝕋𝑣superscriptsubscript𝑞𝑣1𝑠2superscriptsubscript𝑞𝑣1𝑠2subscript1subscript𝕂𝑣𝜑subscriptchsubscript𝑍𝑣subscript𝕂𝑣[R({\mathbb{T}}_{v}-(q_{v}^{(1+s)/2}+q_{v}^{(1-s)/2})\,1_{{\mathbb{K}}_{v}})]\,\varphi={\rm ch}_{Z_{v}{\mathbb{K}}_{v}}.

  • (c)

    φ(g)=O(1)𝜑𝑔𝑂1\varphi(g)=O(1), gGv𝑔subscript𝐺𝑣g\in G_{v}.

Here 𝕋v=vol(𝐊v;dgv)1ch𝐊v[ϖv001]𝐊v{\mathbb{T}}_{v}={\operatorname{vol}}({\mathbf{K}}_{v};dg_{v})^{-1}{\rm ch}_{{\mathbf{K}}_{v}[\begin{smallmatrix}\varpi_{v}&0\\ 0&1\end{smallmatrix}]{\mathbf{K}}_{v}} and 1𝕂v=vol(𝕂v;dgv)1ch𝕂v1_{{\mathbb{K}}_{v}}={\operatorname{vol}}({\mathbb{K}}_{v};{{d}}g_{v})^{-1}{\rm ch}_{{\mathbb{K}}_{v}}. We denote this function φ𝜑\varphi by Φv(s;)subscriptΦ𝑣𝑠\Phi_{v}(s;-). By the Cartan decomposition, it is explicitly written as

Φv(s;[ϖvn100ϖvn2])=(qv(s+1)/2qv(s+1)/2)1qv(s+1)2(n1n2),(n1,n2,n1n2).\displaystyle\Phi_{v}\left(s;\left[\begin{smallmatrix}\varpi_{v}^{n_{1}}&0\\ 0&\varpi_{v}^{n_{2}}\end{smallmatrix}\right]\right)=(q_{v}^{-(s+1)/2}-q_{v}^{(s+1)/2})^{-1}\,q_{v}^{-\frac{(s+1)}{2}(n_{1}-n_{2})},\quad(n_{1},n_{2}\in{\mathbb{Z}},\,n_{1}\geqslant n_{2}).

We also have

(2.3) Φv(s;g)=(qv(s+1)/2qv(s+1)/2)1{|detg|v1max(|a|v,|b|v,|c|v,|d|v)2}(s+1)/2forg=[abcd].\displaystyle\Phi_{v}(s;g)=(q_{v}^{-(s+1)/2}-q_{v}^{(s+1)/2})^{-1}\,\{|\det g|_{v}^{-1}\,\max(|a|_{v},|b|_{v},|c|_{v},|d|_{v})^{2}\}^{-(s+1)/2}\quad\text{for}\,g=\left[\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right].

Let U𝑈U be a compact set of the half plane Re(s)>1Re𝑠1\operatorname{Re}(s)>1. Then, from the formula above,

(2.4) |Φv(s;gv)|Ugvv(Re(s)+1)/2,gvGv,sU.formulae-sequencesubscriptmuch-less-than𝑈subscriptΦ𝑣𝑠subscript𝑔𝑣superscriptsubscriptnormsubscript𝑔𝑣𝑣Re𝑠12formulae-sequencesubscript𝑔𝑣subscript𝐺𝑣𝑠𝑈\displaystyle|\Phi_{v}(s;g_{v})|\ll_{U}\|g_{v}\|_{v}^{-(\operatorname{Re}(s)+1)/2},\quad g_{v}\in G_{v},\,s\in U.
Lemma 2.4.

Let φ:Gv:𝜑subscript𝐺𝑣\varphi:G_{v}\rightarrow\mathbb{C} be a smooth function such that φ(zgk)=φ(g)𝜑𝑧𝑔𝑘𝜑𝑔\varphi(zgk)=\varphi(g) for all gGv𝑔subscript𝐺𝑣g\in G_{v}, zZv𝑧subscript𝑍𝑣z\in Z_{v} and k𝐊v𝑘subscript𝐊𝑣k\in{\mathbf{K}}_{v}. Then we have

Zv\GvΦv(s;gv)[R(𝕋v(qv(1+s)/2+qv(1s)/2) 1𝕂v)φ](gv)𝑑gv=vol(Zv\Zv𝐊v)φ(12)subscript\subscript𝑍𝑣subscript𝐺𝑣subscriptΦ𝑣𝑠subscript𝑔𝑣delimited-[]𝑅subscript𝕋𝑣superscriptsubscript𝑞𝑣1𝑠2superscriptsubscript𝑞𝑣1𝑠2subscript1subscript𝕂𝑣𝜑subscript𝑔𝑣differential-dsubscript𝑔𝑣vol\subscript𝑍𝑣subscript𝑍𝑣subscript𝐊𝑣𝜑subscript12\displaystyle\int_{Z_{v}\backslash G_{v}}\Phi_{v}(s;g_{v})\,[R({\mathbb{T}}_{v}-(q_{v}^{(1+s)/2}+q_{v}^{(1-s)/2})\,{\mathbb{1}}_{{\mathbb{K}}_{v}})\,\varphi](g_{v})\,{{d}}g_{v}={\operatorname{vol}}(Z_{v}\backslash Z_{v}{\mathbf{K}}_{v})\varphi(1_{2})

as long as the integral of the left-hand side is absolutely convergent.

2.4. The kernel functions

Let 𝔫𝔬𝔫𝔬{\mathfrak{n}}\subset\mathfrak{o} be a non-zero ideal and l=(lv)vΣ𝑙subscriptsubscript𝑙𝑣𝑣subscriptΣl=(l_{v})_{v\in\Sigma_{\infty}} an element of (2)Σsuperscript2subscriptΣ(2{\mathbb{N}})^{\Sigma_{\infty}} such that lv4subscript𝑙𝑣4l_{v}\geqslant 4 for all vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Let SΣfin𝑆subscriptΣfinS\subset\Sigma_{\rm fin} be a finite set. For any 𝕤=(sv)vS𝔛S𝕤subscriptsubscript𝑠𝑣𝑣𝑆subscript𝔛𝑆{\mathbb{s}}=(s_{v})_{v\in S}\in{\mathfrak{X}}_{S}, set

Φl(𝔫|𝕤;g)={vΣΦvlv(gv)}{vSΦv(sv;gv)}{vΣfinSΦ𝔫,v(gv)},g=(gv)G𝔸,formulae-sequencesuperscriptΦ𝑙conditional𝔫𝕤𝑔subscriptproduct𝑣subscriptΣsuperscriptsubscriptΦ𝑣subscript𝑙𝑣subscript𝑔𝑣subscriptproduct𝑣𝑆subscriptΦ𝑣subscript𝑠𝑣subscript𝑔𝑣subscriptproduct𝑣subscriptΣfin𝑆subscriptΦ𝔫𝑣subscript𝑔𝑣𝑔subscript𝑔𝑣subscript𝐺𝔸\displaystyle\Phi^{l}({\mathfrak{n}}|{\mathbb{s}};g)=\{\prod_{v\in\Sigma_{\infty}}\Phi_{v}^{l_{v}}(g_{v})\}\,\{\prod_{v\in S}\Phi_{v}(s_{v};g_{v})\}\,\{\prod_{v\in\Sigma_{\rm fin}-S}\Phi_{{\mathfrak{n}},v}(g_{v})\},\quad g=(g_{v})\in G_{\mathbb{A}},

where Φ𝔫,v=chZv𝕂0(𝔫𝔬v)subscriptΦ𝔫𝑣subscriptchsubscript𝑍𝑣subscript𝕂0𝔫subscript𝔬𝑣\Phi_{{\mathfrak{n}},v}={\rm ch}_{Z_{v}{\mathbb{K}}_{0}({\mathfrak{n}}\mathfrak{o}_{v})}. Define a function on G𝔸×G𝔸subscript𝐺𝔸subscript𝐺𝔸G_{\mathbb{A}}\times G_{\mathbb{A}} by

(2.5) 𝚽l(𝔫|𝕤;g,h)superscript𝚽𝑙conditional𝔫𝕤𝑔\displaystyle{\bf\Phi}^{l}({\mathfrak{n}}|{\mathbb{s}};g,h) =γZF\GFΦl(𝔫|𝕤;g1γh),g,hG𝔸.formulae-sequenceabsentsubscript𝛾\subscript𝑍𝐹subscript𝐺𝐹superscriptΦ𝑙conditional𝔫𝕤superscript𝑔1𝛾𝑔subscript𝐺𝔸\displaystyle=\sum_{\gamma\in Z_{F}\backslash G_{F}}\Phi^{l}({\mathfrak{n}}|{\mathbb{s}};g^{-1}\gamma h),\quad g,h\in G_{\mathbb{A}}.
Lemma 2.5.

Let 𝒰𝒰{\mathcal{U}} be a compact subset of {𝐬𝔛S|Re(sv)>1(vS)}conditional-set𝐬subscript𝔛𝑆Resubscript𝑠𝑣1for-all𝑣𝑆\{{\mathbf{s}}\in{\mathfrak{X}}_{S}|\,\operatorname{Re}(s_{v})>1\,(\forall v\in S)\,\}. Then the series (2.5) converges absolutely and

γZF\GF|Φl(𝔫|𝕤;g1γh)|𝒰y(g)m/2,(g,h)𝔖1×G𝔸,𝕤𝒰\displaystyle\sum_{\gamma\in Z_{F}\backslash G_{F}}|\Phi^{l}({\mathfrak{n}}|{\mathbb{s}};g^{-1}\gamma h)|\ll_{{\mathcal{U}}}y(g)^{m/2},\qquad(g,h)\in{\mathfrak{S}}^{1}\times G_{\mathbb{A}},\,{\mathbb{s}}\in{\mathcal{U}}

with m𝑚m being the minimum of the set {lv|vΣ}{Re(sv)+1|𝕤𝒰}conditional-setsubscript𝑙𝑣𝑣subscriptΣconditional-setResubscript𝑠𝑣1𝕤𝒰\{l_{v}|\,v\in\Sigma_{\infty}\}\cup\{\operatorname{Re}(s_{v})+1|\,{\mathbb{s}}\in{\mathcal{U}}\,\}.

Proof.

This follows from (2.2), (2.4) and Proposition 2.2. ∎

Let 𝒜cusp(l,𝔫)subscript𝒜cusp𝑙𝔫{\mathcal{A}}_{\rm cusp}(l,{\mathfrak{n}}) denote the space of all those functions φC(Z𝔸GF\G𝔸)[τl]𝕂0(𝔫)𝜑superscript𝐶\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸superscriptdelimited-[]subscript𝜏𝑙subscript𝕂0𝔫\varphi\in C^{\infty}(Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}})[\tau_{l}]^{{\mathbb{K}}_{0}({\mathfrak{n}})} square integrable on Z𝔸GF\G𝔸\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}} and R(Wv)φ=0𝑅subscript𝑊𝑣𝜑0R(W_{v})\,\varphi=0 for all vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Then 𝒜cusp(l,𝔫)subscript𝒜cusp𝑙𝔫{\mathcal{A}}_{\rm cusp}(l,{\mathfrak{n}}) is contained in Lcusp2(Z𝔸GF\G𝔸)superscriptsubscript𝐿cusp2\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸L_{\rm cusp}^{2}(Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}) by Wallach’s criterion [31, Theorem 4.3]. It is known that the space 𝒜cusp(l,𝔫)subscript𝒜cusp𝑙𝔫{\mathcal{A}}_{\rm cusp}(l,{\mathfrak{n}}) is finite dimensional.

Proposition 2.6.

Suppose 𝕤𝔛S=vS/4πi(logqv)1𝕤subscript𝔛𝑆subscriptproduct𝑣𝑆4𝜋𝑖superscriptsubscript𝑞𝑣1{\mathbb{s}}\in{\mathfrak{X}}_{S}=\prod_{v\in S}\mathbb{C}/4\pi i(\log q_{v})^{-1}\mathbb{Z} with Re(sv)>1Resubscript𝑠𝑣1\operatorname{Re}(s_{v})>1 for all vS𝑣𝑆v\in S and lv4subscript𝑙𝑣4l_{v}\geqslant 4 for all vΣ𝑣subscriptΣv\in\Sigma_{\infty}.

  • (1)

    For any gG𝔸𝑔subscript𝐺𝔸g\in G_{\mathbb{A}}, the function h𝚽l(𝔫|𝕤;g,h)maps-tosuperscript𝚽𝑙conditional𝔫𝕤𝑔h\mapsto{\bf\Phi}^{l}({\mathfrak{n}}|{\mathbb{s}};g,h) belongs to the space 𝒜cusp(l,𝔫)subscript𝒜cusp𝑙𝔫{\mathcal{A}}_{\rm cusp}(l,{\mathfrak{n}}).

  • (2)

    Let (l,𝔫)𝑙𝔫{\mathcal{B}}(l,{\mathfrak{n}}) be an orthonormal basis of the space 𝒜cusp(l,𝔫)subscript𝒜cusp𝑙𝔫{\mathcal{A}}_{\rm cusp}(l,{\mathfrak{n}}) endowed with the induced L2superscript𝐿2L^{2}-inner product from L2(Z𝔸GF\G𝔸)superscript𝐿2\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸L^{2}(Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}). Let (νv(φ))vS𝔛Ssubscriptsubscript𝜈𝑣𝜑𝑣𝑆subscript𝔛𝑆(\nu_{v}(\varphi))_{v\in S}\in{\mathfrak{X}}_{S} for each φ(l,𝔫)𝜑𝑙𝔫\varphi\in{\mathcal{B}}(l,{\mathfrak{n}}) be a family such that

    (2.6) R(𝕋v)φ=(qv(1+νv(φ))/2+qv(1νv(φ))/2)φ𝑅subscript𝕋𝑣𝜑superscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜑2superscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜑2𝜑\displaystyle R({\mathbb{T}}_{v})\,\varphi=(q_{v}^{(1+\nu_{v}(\varphi))/2}+q_{v}^{(1-\nu_{v}(\varphi))/2})\,\varphi

    for all vS𝑣𝑆v\in S. Then, 𝚽l(𝔫|𝐬;g,h)superscript𝚽𝑙conditional𝔫𝐬𝑔{\bf\Phi}^{l}({\mathfrak{n}}|{\mathbf{s}};g,h) with g,hG𝔸𝑔subscript𝐺𝔸g,h\in G_{\mathbb{A}} equals

    (2.7) C(l,𝔫)φ(l,𝔫)φ(g)¯φ(h)vS{(qv(1+νv(φ))/2+qv(1νv(φ))/2)(qv(1+sv)/2+qv(1sv)/2)}1,𝐶𝑙𝔫subscript𝜑𝑙𝔫¯𝜑𝑔𝜑subscriptproduct𝑣𝑆superscriptsuperscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜑2superscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜑2superscriptsubscript𝑞𝑣1subscript𝑠𝑣2superscriptsubscript𝑞𝑣1subscript𝑠𝑣21\displaystyle C(l,{\mathfrak{n}})\,\sum_{\varphi\in{\mathcal{B}}(l,{\mathfrak{n}})}{{\overline{\varphi(g)}}\,\varphi(h)}{\prod_{v\in S}\{(q_{v}^{(1+\nu_{v}(\varphi))/2}+q_{v}^{(1-\nu_{v}(\varphi))/2})-(q_{v}^{(1+s_{v})/2}+q_{v}^{(1-s_{v})/2})\}^{-1}},

    where C(l,𝔫)𝐶𝑙𝔫C(l,{\mathfrak{n}}) is the constant (1.5).

Proof.

The assertion (1) follows from Proposition 2.2 (ii) together with the properties (a) and (b) of ΦvlvsuperscriptsubscriptΦ𝑣subscript𝑙𝑣\Phi_{v}^{l_{v}} in § 2.2.

(2) To simplify notation, we write Φ(h)Φ{\Phi}(h) in place of Φl(𝔫|𝐬;h)superscriptΦ𝑙conditional𝔫𝐬{\Phi}^{l}({\mathfrak{n}}|{\mathbf{s}};h) in this proof. For any φ(l,𝔫)𝜑𝑙𝔫\varphi\in{\mathcal{B}}(l,{\mathfrak{n}}), from the well-known computation ([5]), we have

𝚽l(𝔫|𝐬;g,)|φL2=Z𝔸\G𝔸Φ(h)φ¯(gh)𝑑h.subscriptdelimited-⟨⟩conditionalsuperscript𝚽𝑙conditional𝔫𝐬𝑔𝜑superscript𝐿2subscript\subscript𝑍𝔸subscript𝐺𝔸Φ¯𝜑𝑔differential-d\displaystyle\langle{\bf\Phi}^{l}({\mathfrak{n}}|{\mathbf{s}};g,\bullet)|\varphi\rangle_{L^{2}}=\textstyle{\int}_{Z_{\mathbb{A}}\backslash G_{\mathbb{A}}}{\Phi}(h)\,\bar{\varphi}(gh)\,{{d}}h.

Let us write a general element hG𝔸subscript𝐺𝔸h\in G_{\mathbb{A}} as h=hhShS,subscriptsubscript𝑆superscript𝑆h=h_{\infty}h_{S}h^{S,\infty} with hGsubscriptsubscript𝐺h_{\infty}\in G_{\infty}, hSGS=vSGvsubscript𝑆subscript𝐺𝑆subscriptproduct𝑣𝑆subscript𝐺𝑣h_{S}\in G_{S}=\prod_{v\in S}G_{v} and hS,vΣfinSGvsuperscript𝑆subscriptproduct𝑣subscriptΣfin𝑆subscript𝐺𝑣h^{S,\infty}\in\prod_{v\in\Sigma_{\rm fin}-S}G_{v}. Suppose Φ(h)0Φ0{\Phi}(h)\not=0. Then hS,vΣfinS𝕂0(𝔫𝔬v)superscript𝑆subscriptproduct𝑣subscriptΣfin𝑆subscript𝕂0𝔫subscript𝔬𝑣h^{S,\infty}\in\prod_{v\in\Sigma_{\rm fin}-S}{\mathbb{K}}_{0}({\mathfrak{n}}\mathfrak{o}_{v}), and thus φ(gh)=φ(ghhS)𝜑𝑔𝜑𝑔subscriptsubscript𝑆\varphi(gh)=\varphi(gh_{\infty}h_{S}). Hence the hS,superscript𝑆h^{S,\infty}-integral yields the volume factor

vΣfinSvol(Zv\Zv𝕂0(𝔫𝔬v);dhv)=[𝕂fin:𝕂0(𝔫)]1vΣfinSvol(Zv\Zv𝕂v;dhv).\prod_{v\in\Sigma_{\rm fin}-S}{\operatorname{vol}}(Z_{v}\backslash Z_{v}{\mathbb{K}}_{0}({\mathfrak{n}}\mathfrak{o}_{v});{{d}}h_{v})=[{\mathbb{K}}_{\rm fin}:{\mathbb{K}}_{0}({\mathfrak{n}})]^{-1}\,\prod_{v\in\Sigma_{\rm fin}-S}{\operatorname{vol}}(Z_{v}\backslash Z_{v}{\mathbb{K}}_{v};{{d}}h_{v}).

The hsubscripth_{\infty}-integral and hSsubscript𝑆h_{S}-integral are computed by Lemmas 2.3 and 2.4, respectively. To complete the proof, we use vol(Zv\Zv𝕂v)=qvdvvol\subscript𝑍𝑣subscript𝑍𝑣subscript𝕂𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣{\operatorname{vol}}(Z_{v}\backslash Z_{v}{\mathbb{K}}_{v})=q_{v}^{-d_{v}} for vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and the relation νv(φ¯)=±νv(φ)subscript𝜈𝑣¯𝜑plus-or-minussubscript𝜈𝑣𝜑\nu_{v}(\bar{\varphi})=\pm\nu_{v}(\varphi) in 𝔛v=/4πi(logqv)1subscript𝔛𝑣4𝜋𝑖superscriptsubscript𝑞𝑣1{\mathfrak{X}}_{v}=\mathbb{C}/4\pi i(\log q_{v})^{-1}\mathbb{Z} for vS𝑣𝑆v\in S which results from the self-adjointness of 𝕋vsubscript𝕋𝑣{\mathbb{T}}_{v}. ∎

Remark : Let π𝜋\pi be the cuspidal automorphic representation generated by φ(l,𝔫)𝜑𝑙𝔫\varphi\in{\mathcal{B}}(l,{\mathfrak{n}}). Then, νv(φ)=±νv(π)subscript𝜈𝑣𝜑plus-or-minussubscript𝜈𝑣𝜋\nu_{v}(\varphi)=\pm\nu_{v}(\pi) for any vS𝑣𝑆v\in S. It is known to be purely imaginary by Blasius [1]. We do not need this fact in this paper.

3. Smoothed convolution product: The spectral side

3.1. Smoothed Eisenstein series

Let

E(z;g)=γBF\GFy(γg)z+12,Re(z)>1,gG𝔸formulae-sequence𝐸𝑧𝑔subscript𝛾\subscript𝐵𝐹subscript𝐺𝐹𝑦superscript𝛾𝑔𝑧12formulae-sequenceRe𝑧1𝑔subscript𝐺𝔸E(z;g)=\sum_{\gamma\in B_{F}\backslash G_{F}}y(\gamma g)^{\frac{z+1}{2}},\quad\operatorname{Re}(z)>1,\,g\in G_{\mathbb{A}}

be the 𝕂𝕂{\mathbb{K}}-spherical Eisenstein series on G𝔸subscript𝐺𝔸G_{\mathbb{A}}. As a function in z𝑧z, it has a meromorphic continuation to {\mathbb{C}}, holomorphic on Re(z)0Re𝑧0\operatorname{Re}(z)\geqslant 0 away from the simple pole at z=1𝑧1z=1 and satisfying the functional equation E(z;g)=E(z;g)superscript𝐸𝑧𝑔superscript𝐸𝑧𝑔E^{*}(-z;g)=E^{*}(z;g), where

E(z;g)=ΛF(z+1)E(z;g),ΛF(z)=DFz/2ζF(z).formulae-sequencesuperscript𝐸𝑧𝑔subscriptΛ𝐹𝑧1𝐸𝑧𝑔subscriptΛ𝐹𝑧superscriptsubscript𝐷𝐹𝑧2subscript𝜁𝐹𝑧E^{*}(z;g)=\Lambda_{F}(z+1)\,E(z;g),\quad\Lambda_{F}(z)=D_{F}^{z/2}\zeta_{F}(z).

We have the functional equation ΛF(1z)=ΛF(z)subscriptΛ𝐹1𝑧subscriptΛ𝐹𝑧\Lambda_{F}(1-z)=\Lambda_{F}(z) and the Fourier expansion

(3.1) E(z;g)superscript𝐸𝑧𝑔\displaystyle E^{*}(z;g) =ΛF(z)y(g)(1+z)/2+ΛF(z)y(g)(1z)/2+ΛF(z)aF×Wψ(z;[a001]g),absentsubscriptΛ𝐹𝑧𝑦superscript𝑔1𝑧2subscriptΛ𝐹𝑧𝑦superscript𝑔1𝑧2subscriptΛ𝐹𝑧subscript𝑎superscript𝐹subscript𝑊𝜓𝑧delimited-[]𝑎001𝑔\displaystyle=\Lambda_{F}(-z)\,y(g)^{(1+z)/2}+\Lambda_{F}(z)\,y(g)^{(1-z)/2}+\Lambda_{F}(-z)\,\sum_{a\in F^{\times}}W_{\psi}\left(z;\left[\begin{smallmatrix}a&0\\ 0&1\end{smallmatrix}\right]g\right),

where Wψ(z)subscript𝑊𝜓𝑧W_{\psi}(z) is the global Whittaker function defined as

Wψ(z;g)=N𝔸y(w0[1x01]g)(z+1)/2ψ(x)𝑑x,gG𝔸.formulae-sequencesubscript𝑊𝜓𝑧𝑔subscriptsubscript𝑁𝔸𝑦superscriptsubscript𝑤0delimited-[]1𝑥01𝑔𝑧12𝜓𝑥differential-d𝑥𝑔subscript𝐺𝔸W_{\psi}(z;g)=\int_{N_{\mathbb{A}}}y\left(w_{0}\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]g\right)^{(z+1)/2}\,\psi(-x){{d}}x,\quad g\in G_{\mathbb{A}}.

We have the product formula Wψ(z;g)=vΣFWv(z;gv)subscript𝑊𝜓𝑧𝑔subscriptproduct𝑣subscriptΣ𝐹subscript𝑊𝑣𝑧subscript𝑔𝑣W_{\psi}(z;g)=\prod_{v\in\Sigma_{F}}W_{v}(z;g_{v}), where Wv(z;gv)subscript𝑊𝑣𝑧subscript𝑔𝑣W_{v}(z;g_{v}) is the 𝕂vsubscript𝕂𝑣{\mathbb{K}}_{v}-invariant Whittaker function on Gvsubscript𝐺𝑣G_{v} determined by

(3.2) Wv(z;[t001])=ζFv(z+1)1δ(ϖvdvt𝔬v)|ϖvdvt|v1/2|ϖvdv+1t|vz/2|ϖvdv+1t|vz/2qvz/2qvz/2,tFv×formulae-sequencesubscript𝑊𝑣𝑧delimited-[]𝑡001subscript𝜁subscript𝐹𝑣superscript𝑧11𝛿superscriptsubscriptitalic-ϖ𝑣subscript𝑑𝑣𝑡subscript𝔬𝑣superscriptsubscriptsuperscriptsubscriptitalic-ϖ𝑣subscript𝑑𝑣𝑡𝑣12superscriptsubscriptsuperscriptsubscriptitalic-ϖ𝑣subscript𝑑𝑣1𝑡𝑣𝑧2superscriptsubscriptsuperscriptsubscriptitalic-ϖ𝑣subscript𝑑𝑣1𝑡𝑣𝑧2superscriptsubscript𝑞𝑣𝑧2superscriptsubscript𝑞𝑣𝑧2𝑡superscriptsubscript𝐹𝑣\displaystyle W_{v}\left(z;\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]\right)=\zeta_{F_{v}}(z+1)^{-1}\delta(\varpi_{v}^{d_{v}}\,t\in\mathfrak{o}_{v})\,|\varpi_{v}^{d_{v}}t|_{v}^{1/2}\,\tfrac{|\varpi_{v}^{d_{v}+1}t|_{v}^{z/2}-|\varpi_{v}^{d_{v}+1}t|_{v}^{-z/2}}{q_{v}^{-z/2}-q_{v}^{z/2}},\quad t\in F_{v}^{\times}

if vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and by

(3.3) Wv(z;[t001])=ζFv(z+1)1 2|t|v1/2Kz/2(2π|t|v),tFv×formulae-sequencesubscript𝑊𝑣𝑧delimited-[]𝑡001subscript𝜁subscript𝐹𝑣superscript𝑧112superscriptsubscript𝑡𝑣12subscript𝐾𝑧22𝜋subscript𝑡𝑣𝑡superscriptsubscript𝐹𝑣\displaystyle W_{v}\left(z;\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]\right)=\zeta_{F_{v}}(z+1)^{-1}\,2\,|t|_{v}^{1/2}\,K_{z/2}(2\pi|t|_{v}),\quad t\in F_{v}^{\times}

if vΣ𝑣subscriptΣv\in\Sigma_{\infty}. We need the uniform estimate of the Eisenstein series:

Lemma 3.1.

For any 0<σ1<σ20subscript𝜎1subscript𝜎20<\sigma_{1}<\sigma_{2}, there exists m>0𝑚0m>0 such that, for any element D𝐷D of the universal enveloping algebra of G(F)G𝔸1𝐺subscript𝐹superscriptsubscript𝐺𝔸1G(F_{\infty})\cap G_{\mathbb{A}}^{1}, it holds that

|(z1)R(D)E(z;g)|Dy(g)m,Re(z)[σ1,σ2],g𝔖1.formulae-sequencesubscriptmuch-less-than𝐷𝑧1𝑅𝐷superscript𝐸𝑧𝑔𝑦superscript𝑔𝑚formulae-sequenceRe𝑧subscript𝜎1subscript𝜎2𝑔superscript𝔖1|(z-1)R(D)\,E^{*}(z;g)|\ll_{D}y(g)^{m},\quad\operatorname{Re}(z)\in[\sigma_{1},\sigma_{2}],\,g\in{\mathfrak{S}}^{1}.
Proof.

This follows from the materials proved in [11, §19]. Here is a brief indication of the proof. Let ΦΦ\Phi be a Schwartz-Bruhat function on 𝔸2superscript𝔸2{\mathbb{A}}^{2} defined as vΣFΦv0subscripttensor-product𝑣subscriptΣ𝐹absentsubscriptsuperscriptΦ0𝑣\otimes_{v\in\Sigma_{F}}\Phi^{0}_{v} with Φv0=ch𝔬v𝔬vsubscriptsuperscriptΦ0𝑣subscriptchdirect-sumsubscript𝔬𝑣subscript𝔬𝑣\Phi^{0}_{v}={\rm ch}_{\mathfrak{o}_{v}\oplus\mathfrak{o}_{v}} for vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and Φv0(x,y)=eπ(x2+y2)superscriptsubscriptΦ𝑣0𝑥𝑦superscript𝑒𝜋superscript𝑥2superscript𝑦2\Phi_{v}^{0}(x,y)=e^{-\pi(x^{2}+y^{2})} for vΣ𝑣subscriptΣv\in\Sigma_{\infty}. By the multiplication from the right, G(F)𝐺subscript𝐹G(F_{\infty}) acts on 𝔸2superscript𝔸2{\mathbb{A}}^{2} and hence on the space of Schwartz-Bruhat functions; by the induced action of the universal enveloping algebra of G(F)𝐺subscript𝐹G(F_{\infty}), we can define the derivative DΦ𝐷ΦD\Phi of ΦΦ\Phi. Let D𝐷D be as in the lemma. Then DFz/2E(g.DΦ,||𝔸z/2,||𝔸z/2)D_{F}^{z/2}E(g.D\Phi,|\,|_{\mathbb{A}}^{z/2},|\,|_{\mathbb{A}}^{-z/2}) in [11, §19] coincides with our R(D)E(z,g)𝑅𝐷superscript𝐸𝑧𝑔R(D)E^{*}(z,g) up to a non-zero constant multiple; this is checked by a computation on their absolute-convergence region Re(z)>1Re𝑧1\operatorname{Re}(z)>1. The desired bound follows from a similar bound of θ0(||𝔸(z+1)/2,g.Φ)\theta^{0}(|\,|_{\mathbb{A}}^{(z+1)/2},g.\Phi) by the formula [11, (19.4)]. ∎

Let 1subscript1{\mathcal{B}}_{1} be the space of entire functions β(z)𝛽𝑧\beta(z) such that β(0)=β(±1)=β(±1)=0𝛽0𝛽plus-or-minus1superscript𝛽plus-or-minus10\beta(0)=\beta(\pm 1)=\beta^{\prime}(\pm 1)=0 and such that on any interval [σ1,σ2]subscript𝜎1subscript𝜎2[\sigma_{1},\sigma_{2}]\subset{\mathbb{R}} and N>0𝑁0N>0,

|β(z)|(1+|Im(z)|)N,Re(z)[σ1,σ2].formulae-sequencemuch-less-than𝛽𝑧superscript1Im𝑧𝑁Re𝑧subscript𝜎1subscript𝜎2|\beta(z)|\ll(1+|{\operatorname{Im}}(z)|)^{-N},\quad\operatorname{Re}(z)\in[\sigma_{1},\sigma_{2}].

For β1𝛽subscript1\beta\in{\mathcal{B}}_{1} and σ>0𝜎0\sigma>0, we set

β(g)=Lσβ(z)E(z;g)𝑑z,gG𝔸.formulae-sequencesuperscriptsubscript𝛽𝑔subscriptsubscript𝐿𝜎𝛽𝑧superscript𝐸𝑧𝑔differential-d𝑧𝑔subscript𝐺𝔸\displaystyle{\mathcal{E}}_{\beta}^{*}(g)=\int_{L_{\sigma}}\beta(z)\,E^{*}(z;g)\,{{d}}z,\quad g\in G_{\mathbb{A}}.
Proposition 3.2.

The contour integral β(g)superscriptsubscript𝛽𝑔{\mathcal{E}}_{\beta}^{*}(g) converges absolutely and is independent of the choice of a contour Lσ(σ>0)subscript𝐿𝜎𝜎0L_{\sigma}\,(\sigma>0). For any N>0𝑁0N>0,

|β(g)|y(g)N,g𝔖1.formulae-sequencemuch-less-thansuperscriptsubscript𝛽𝑔𝑦superscript𝑔𝑁𝑔superscript𝔖1\displaystyle|{\mathcal{E}}_{\beta}^{*}(g)|\ll y(g)^{-N},\quad g\in{\mathfrak{S}}^{1}.
Proof.

Due to β(1)=0𝛽10\beta(1)=0, the function β(z)E(z;g)𝛽𝑧superscript𝐸𝑧𝑔\beta(z)E^{*}(z;g) is holomorphic on Re(z)>0Re𝑧0\operatorname{Re}(z)>0. Thus the first two assertions follow from Lemma 3.1. From Lemma 3.1, by means of [17, Lemma I.2.10], we can deduce the following estimate for the non-constant term of the Eisenstein series ENC(z;g)=E(z;g){ΛF(z)y(g)z+12+ΛF(z)y(g)z+12}superscriptsubscript𝐸NC𝑧𝑔superscript𝐸𝑧𝑔subscriptΛ𝐹𝑧𝑦superscript𝑔𝑧12subscriptΛ𝐹𝑧𝑦superscript𝑔𝑧12E_{\rm NC}^{*}(z;g)=E^{*}(z;g)-\{\Lambda_{F}(-z)y(g)^{\frac{z+1}{2}}+\Lambda_{F}(z)y(g)^{\frac{-z+1}{2}}\}:

(3.4) |ENC(z;g)|ny(g)n,z𝒯δ,g𝔖1,formulae-sequencesubscriptmuch-less-than𝑛subscript𝐸NC𝑧𝑔𝑦superscript𝑔𝑛formulae-sequence𝑧subscript𝒯𝛿𝑔superscript𝔖1\displaystyle|E_{\rm NC}(z;g)|\ll_{n}y(g)^{-n},\quad z\in{\mathcal{T}}_{\delta},\,g\in{\mathfrak{S}}^{1},

for arbitrary n>0𝑛0n>0. To argue, we write β(g)superscriptsubscript𝛽𝑔{\mathcal{E}}_{\beta}^{*}(g) as a sum of the following three terms:

I+(g)subscript𝐼𝑔\displaystyle I_{+}(g) =L3/2β(z)ΛF(z)y(g)z+12𝑑z,I(g)=L3/2β(z)ΛF(z)y(g)z+12𝑑z,formulae-sequenceabsentsubscriptsubscript𝐿32𝛽𝑧subscriptΛ𝐹𝑧𝑦superscript𝑔𝑧12differential-d𝑧subscript𝐼𝑔subscriptsubscript𝐿32𝛽𝑧subscriptΛ𝐹𝑧𝑦superscript𝑔𝑧12differential-d𝑧\displaystyle=\textstyle{\int}_{L_{3/2}}\beta(z)\Lambda_{F}(-z)y(g)^{\frac{z+1}{2}}{{d}}z,\quad I_{-}(g)=\textstyle{\int}_{L_{3/2}}\beta(z)\Lambda_{F}(z)y(g)^{\frac{-z+1}{2}}{{d}}z,
INC(g)subscript𝐼NC𝑔\displaystyle I_{\rm NC}(g) =L3/2β(z)ΛF(z)ENC(z;g)𝑑z.absentsubscriptsubscript𝐿32𝛽𝑧subscriptΛ𝐹𝑧subscript𝐸NC𝑧𝑔differential-d𝑧\displaystyle=\textstyle{\int}_{L_{3/2}}\beta(z)\Lambda_{F}(-z)E_{\rm NC}(z;g)\,{{d}}z.

By (3.4), the integral INC(g)subscript𝐼NC𝑔I_{\rm NC}(g) has the majorant y(g)N𝑦superscript𝑔𝑁y(g)^{-N} on 𝔖1superscript𝔖1{\mathfrak{S}}^{1} with an arbitrary large N𝑁N. Due to β(0)=β(±1)=0𝛽0𝛽plus-or-minus10\beta(0)=\beta(\pm 1)=0, the integrands of I±(g)subscript𝐼plus-or-minus𝑔I_{\pm}(g) are holomorphic on {\mathbb{C}}. Thus by shifting the contour, the integrals I±(g)subscript𝐼plus-or-minus𝑔I_{\pm}(g) are also shown to be bounded by y(g)N𝑦superscript𝑔𝑁y(g)^{-N} on 𝔖1superscript𝔖1{\mathfrak{S}}^{1}. ∎

3.2. The smoothed convolution

In this section, we fix 𝐬=(sv)vS𝔛S𝐬subscriptsubscript𝑠𝑣𝑣𝑆subscript𝔛𝑆{\mathbf{s}}=(s_{v})_{v\in S}\in{\mathfrak{X}}_{S} such that Re(sv)>1Resubscript𝑠𝑣1\operatorname{Re}(s_{v})>1 for all vS𝑣𝑆v\in S, and consider

(3.5) 𝕀(𝐬,β)=Z𝔸GF\G𝔸𝚽l(𝔫|𝐬;g,g)β(g)𝑑g,β1.formulae-sequence𝕀𝐬𝛽subscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸superscript𝚽𝑙conditional𝔫𝐬𝑔𝑔superscriptsubscript𝛽𝑔differential-d𝑔𝛽subscript1\displaystyle\mathbb{I}({\mathbf{s}},\beta)=\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}{\bf\Phi}^{l}({\mathfrak{n}}|{\mathbf{s}};g,g){\mathcal{E}}_{\beta}^{*}(g)\,{{d}}g,\quad\beta\in{\mathcal{B}}_{1}.

From Proposition 3.2 and Lemma 2.5, the integral converges absolutely.

Let I^cusp(𝐬,z)subscript^𝐼cusp𝐬𝑧\hat{I}_{\rm{cusp}}({\mathbf{s}},z) be a meromorphic function on {\mathbb{C}} defined as

(3.6) I^cusp(𝐬,z)=subscript^𝐼cusp𝐬𝑧absent\displaystyle\hat{I}_{\rm cusp}({\mathbf{s}},z)= C(l,𝔫)πΠcus(l,𝔫)E(z)(l,𝔫;π)vS{(qv(1+νv(π))/2+qv(1νv(π))/2)(qv(1+sv)/2+qv(1sv)/2)},𝐶𝑙𝔫subscript𝜋subscriptΠcus𝑙𝔫subscriptsuperscript𝐸𝑧𝑙𝔫𝜋subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜋2superscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜋2superscriptsubscript𝑞𝑣1subscript𝑠𝑣2superscriptsubscript𝑞𝑣1subscript𝑠𝑣2\displaystyle C(l,{\mathfrak{n}})\sum_{\pi\in\Pi_{\rm{cus}}(l,{\mathfrak{n}})}\frac{\mathbb{P}_{E^{*}(z)}(l,{\mathfrak{n}};\pi)}{\prod_{v\in S}\{(q_{v}^{(1+\nu_{v}(\pi))/2}+q_{v}^{(1-\nu_{v}(\pi))/2})-(q_{v}^{(1+s_{v})/2}+q_{v}^{(1-s_{v})/2})\}},

where C(l,𝔫)𝐶𝑙𝔫C(l,{\mathfrak{n}}) is the constant (1.5) and

(3.7) E(z)(l,𝔫;π)=φπ(l,𝔫)E(z;)|φφ¯L2subscriptsuperscript𝐸𝑧𝑙𝔫𝜋subscript𝜑subscript𝜋𝑙𝔫subscriptinner-productsuperscript𝐸𝑧𝜑¯𝜑superscript𝐿2\displaystyle\mathbb{P}_{E^{*}(z)}(l,{\mathfrak{n}};\pi)=\sum_{\varphi\in{\mathcal{B}}_{\pi}(l,{\mathfrak{n}})}\langle E^{*}(z;-)|\varphi\,\bar{\varphi}\rangle_{L^{2}}

with π(l,𝔫)subscript𝜋𝑙𝔫{\mathcal{B}}_{\pi}(l,{\mathfrak{n}}) being an orthonormal basis of Vπ[τl]𝕂0(𝔫)subscript𝑉𝜋superscriptdelimited-[]subscript𝜏𝑙subscript𝕂0𝔫V_{\pi}[\tau_{l}]^{{\mathbb{K}}_{0}({\mathfrak{n}})}. Let I𝐼I\subset{\mathbb{R}} be an open interval. A meromorphic function f(z)𝑓𝑧f(z) on the vertical strip Re(z)IRe𝑧𝐼\operatorname{Re}(z)\in I which is holomorphic away from possible poles on the real axis is said to be vertically of moderate growth on the strip Re(z)IRe𝑧𝐼\operatorname{Re}(z)\in I if for any [σ1,σ2]Isubscript𝜎1subscript𝜎2𝐼[\sigma_{1},\sigma_{2}]\subset I there exists N1>0subscript𝑁10N_{1}>0 such that |f(z)|(1+|Im(z)|)N1much-less-than𝑓𝑧superscript1Im𝑧subscript𝑁1|f(z)|\ll(1+|{\operatorname{Im}}(z)|)^{N_{1}} on any 𝒯δ={z|Re(z)[σ1,σ2],|Im(z)|δ}subscript𝒯𝛿conditional-set𝑧formulae-sequenceRe𝑧subscript𝜎1subscript𝜎2Im𝑧𝛿{\mathcal{T}}_{\delta}=\{z\in{\mathbb{C}}|\operatorname{Re}(z)\in[\sigma_{1},\sigma_{2}],\,|{\operatorname{Im}}(z)|\geqslant\delta\}. For example, Lemma 3.1 shows that the functions zE(z)|φφ¯L2maps-to𝑧subscriptinner-productsuperscript𝐸𝑧𝜑¯𝜑superscript𝐿2z\mapsto\langle E^{*}(z)|\varphi\,\bar{\varphi}\rangle_{L^{2}} with φπ(l,𝔫)𝜑subscript𝜋𝑙𝔫\varphi\in{\mathcal{B}}_{\pi}(l,{\mathfrak{n}}) are vertically of moderate growth on {\mathbb{C}}. Since Πcus(l,𝔫)subscriptΠcus𝑙𝔫\Pi_{\rm{cus}}(l,{\mathfrak{n}}) and π(l,𝔫)subscript𝜋𝑙𝔫{\mathcal{B}}_{\pi}(l,{\mathfrak{n}}) are finite sets, we see that the function I^cusp(𝐬,z)subscript^𝐼cusp𝐬𝑧\hat{I}_{\rm{cusp}}({\mathbf{s}},z) is also vertically of moderate growth on {\mathbb{C}} and is holomorphic away from the possible simple poles at z=0,±1𝑧0plus-or-minus1z=0,\pm 1. By the spectral expansion given in Proposition 2.6 (2), we have

(3.8) 𝕀(𝐬,β)=𝕀𝐬𝛽absent\displaystyle\mathbb{I}({\mathbf{s}},\beta)= Lσβ(z)I^cusp(𝐬,z)𝑑z(σ)subscriptsubscript𝐿𝜎𝛽𝑧subscript^𝐼cusp𝐬𝑧differential-d𝑧𝜎\displaystyle\int_{L_{\sigma}}\beta(z)\hat{I}_{\rm cusp}({\mathbf{s}},z)\,dz\quad(\sigma\in{\mathbb{R}})

for all β1𝛽subscript1\beta\in{\mathcal{B}}_{1}.

Let us calculate (3.7) when 𝔫𝔫{\mathfrak{n}} is square-free. For πΠcus(l,𝔫)𝜋subscriptΠcus𝑙𝔫\pi\in\Pi_{\rm cus}(l,{\mathfrak{n}}), the conductor 𝔣πsubscript𝔣𝜋{\mathfrak{f}}_{\pi} of π𝜋\pi divides the ideal 𝔫𝔫{\mathfrak{n}}. Recall the basis π(l,𝔫)={φπ,ρL21φπ,ρ}ρΛπ(𝔫)subscript𝜋𝑙𝔫subscriptsuperscriptsubscriptnormsubscript𝜑𝜋𝜌superscript𝐿21subscript𝜑𝜋𝜌𝜌subscriptΛ𝜋𝔫{\mathcal{B}}_{\pi}(l,{\mathfrak{n}})=\{\|\varphi_{\pi,\rho}\|_{L^{2}}^{-1}\,\varphi_{\pi,\rho}\}_{\rho\in\Lambda_{\pi}({\mathfrak{n}})} constructed in [24] (see also [30]), where the index set Λπ(𝔫)subscriptΛ𝜋𝔫\Lambda_{\pi}({\mathfrak{n}}) consists of all the mappings ρ:ΣF{0,1}:𝜌subscriptΣ𝐹01\rho:\Sigma_{F}\rightarrow\{0,1\} such that ρ(v)=0𝜌𝑣0\rho(v)=0 for all vΣFS(𝔫𝔣π1)𝑣subscriptΣ𝐹𝑆𝔫superscriptsubscript𝔣𝜋1v\in\Sigma_{F}-S({\mathfrak{n}}{\mathfrak{f}}_{\pi}^{-1}) and such that

F\𝔸φπ,ρ([1x01]g)ψ(x)𝑑x=vΣFϕρ(v),v(gv),subscript\𝐹𝔸subscript𝜑𝜋𝜌delimited-[]1𝑥01𝑔𝜓𝑥differential-d𝑥subscriptproduct𝑣subscriptΣ𝐹subscriptitalic-ϕ𝜌𝑣𝑣subscript𝑔𝑣\displaystyle\int_{F\backslash\mathbb{A}}\varphi_{\pi,\rho}([\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}]g)\psi(-x)dx=\prod_{v\in\Sigma_{F}}\phi_{\rho(v),v}(g_{v}),

where ϕj,vsubscriptitalic-ϕ𝑗𝑣\phi_{j,v} are ψvsubscript𝜓𝑣\psi_{v}-Whittaker functions; for vΣ𝑣subscriptΣv\in\Sigma_{\infty}, ϕ0,vsubscriptitalic-ϕ0𝑣\phi_{0,v} is given as

ϕ0,v(t001)=δ(t>0) 2|t|vlv/2e2πt,tFv×,formulae-sequencesubscriptitalic-ϕ0𝑣𝑡001𝛿𝑡02superscriptsubscript𝑡𝑣subscript𝑙𝑣2superscript𝑒2𝜋𝑡𝑡superscriptsubscript𝐹𝑣\phi_{0,v}(\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix})=\delta(t>0)\,2|t|_{v}^{l_{v}/2}\,e^{-2\pi t},\qquad t\in F_{v}^{\times},

and for vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}, ϕ0,vsubscriptitalic-ϕ0𝑣\phi_{0,v} and ϕ1,vsubscriptitalic-ϕ1𝑣\phi_{1,v} will be recalled in the proof of Lemma 3.3. By the unfolding procedure, the Rankin-Selberg integral

E(2s1;)|φπ,ρφπ,ρ¯L2=Z𝔸GF\G𝔸E(2s1;g)φπ,ρ(g)φπ,ρ(g)¯𝑑gsubscriptinner-product𝐸2𝑠1subscript𝜑𝜋𝜌¯subscript𝜑𝜋𝜌superscript𝐿2subscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸𝐸2𝑠1𝑔subscript𝜑𝜋𝜌𝑔¯subscript𝜑𝜋𝜌𝑔differential-d𝑔\langle E(2s-1;-)|\varphi_{\pi,\rho}\,\overline{\varphi_{\pi,\rho}}\rangle_{L^{2}}=\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}E(2s-1;g)\varphi_{\pi,\rho}(g)\overline{\varphi_{\pi,\rho}(g)}dg

for the cusp forms φπ,ρsubscript𝜑𝜋𝜌\varphi_{\pi,\rho} is shown to be decomposed into the product of Zv(s,ϕρ(v),v,ϕρ(v),v)subscript𝑍𝑣𝑠subscriptitalic-ϕ𝜌𝑣𝑣subscriptitalic-ϕ𝜌𝑣𝑣Z_{v}(s,\phi_{\rho(v),v},\phi_{\rho(v),v}) over all vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}, where

(3.9) Zv(s,ϕ,ϕ)=Fv×𝐊vϕ([t001]k)ϕ([t001]k)¯|t|vs1𝑑kd×t,ϕ,ϕ𝒲(πv,ψv)formulae-sequencesubscript𝑍𝑣𝑠italic-ϕsuperscriptitalic-ϕsubscriptsuperscriptsubscript𝐹𝑣subscriptsubscript𝐊𝑣italic-ϕdelimited-[]𝑡001𝑘¯superscriptitalic-ϕdelimited-[]𝑡001𝑘superscriptsubscript𝑡𝑣𝑠1differential-d𝑘superscript𝑑𝑡italic-ϕsuperscriptitalic-ϕ𝒲subscript𝜋𝑣subscript𝜓𝑣\displaystyle Z_{v}(s,\phi,\phi^{\prime})=\int_{F_{v}^{\times}}\int_{{\mathbf{K}}_{v}}\phi([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}]k)\overline{\phi^{\prime}([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}]k)}|t|_{v}^{s-1}dkd^{\times}t,\quad\phi,\phi^{\prime}\in{{\mathcal{W}}}(\pi_{v},\psi_{v})

is the local zeta integral. Recall the quantity Q(πv)𝑄subscript𝜋𝑣Q(\pi_{v}) defined by (1.1). For computations over a finite place v𝑣v, we use the decomposition

(3.10) 𝕂v=𝕂0(𝔭v)ξ𝔬v/𝔭v[1ξ01]w0𝕂0(𝔭v)with w0=[0110].subscript𝕂𝑣subscript𝕂0subscript𝔭𝑣subscript𝜉subscript𝔬𝑣subscript𝔭𝑣delimited-[]1𝜉01subscript𝑤0subscript𝕂0subscript𝔭𝑣with w0=[0110].\displaystyle{\mathbb{K}}_{v}={\mathbb{K}}_{0}({\mathfrak{p}}_{v})\cup\bigcup_{\xi\in\mathfrak{o}_{v}/{\mathfrak{p}}_{v}}\left[\begin{smallmatrix}1&\xi\\ 0&1\end{smallmatrix}\right]w_{0}{\mathbb{K}}_{0}({\mathfrak{p}}_{v})\quad\text{with $w_{0}=\left[\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right]$.}
Lemma 3.3.

For any vΣ𝑣subscriptΣv\in\Sigma_{\infty}, we have

Z(s,ϕ0,v,ϕ0,v)=21lvΓ(s)Γ(2s)L(s,πv;Ad).𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣superscript21subscript𝑙𝑣subscriptΓ𝑠subscriptΓ2𝑠𝐿𝑠subscript𝜋𝑣AdZ(s,\phi_{0,v},\phi_{0,v})=2^{1-l_{v}}\frac{\Gamma_{\mathbb{R}}(s)}{\Gamma_{\mathbb{R}}(2s)}L(s,\pi_{v};{\operatorname{Ad}}).

For vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} with c(πv)=0𝑐subscript𝜋𝑣0c(\pi_{v})=0, we have

(3.11) Z(s,ϕ0,v,ϕ0,v)𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣\displaystyle Z(s,\phi_{0,v},\phi_{0,v}) =qvdv(s3/2)ζF,v(s)ζF,v(2s)L(s,πv;Ad),absentsuperscriptsubscript𝑞𝑣subscript𝑑𝑣𝑠32subscript𝜁𝐹𝑣𝑠subscript𝜁𝐹𝑣2𝑠𝐿𝑠subscript𝜋𝑣Ad\displaystyle=q_{v}^{d_{v}(s-3/2)}\frac{\zeta_{F,v}(s)}{\zeta_{F,v}(2s)}L(s,\pi_{v};{\operatorname{Ad}}),
(3.12) Z(s,ϕ1,v,ϕ1,v)𝑍𝑠subscriptitalic-ϕ1𝑣subscriptitalic-ϕ1𝑣\displaystyle Z(s,\phi_{1,v},\phi_{1,v}) ={qv1s+qvs1+qvQ(πv)2}Z(s,ϕ0,v,ϕ0,v),absentsuperscriptsubscript𝑞𝑣1𝑠superscriptsubscript𝑞𝑣𝑠1subscript𝑞𝑣𝑄superscriptsubscript𝜋𝑣2𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣\displaystyle=\{\frac{q_{v}^{1-s}+q_{v}^{s}}{1+q_{v}}-Q(\pi_{v})^{2}\}Z(s,\phi_{0,v},\phi_{0,v}),

where Q(πv)𝑄subscript𝜋𝑣Q(\pi_{v}) is the number defined in (1.1). For vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} with c(πv)=1𝑐subscript𝜋𝑣1c(\pi_{v})=1, we have

(3.13) Z(s,ϕ0,v,ϕ0,v)=vol(𝐊0(𝔭v))qvs×qvdv(s3/2)ζF,v(s)ζF,v(2s)L(s,πv;Ad).𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣volsubscript𝐊0subscript𝔭𝑣superscriptsubscript𝑞𝑣𝑠superscriptsubscript𝑞𝑣subscript𝑑𝑣𝑠32subscript𝜁𝐹𝑣𝑠subscript𝜁𝐹𝑣2𝑠𝐿𝑠subscript𝜋𝑣Ad\displaystyle Z(s,\phi_{0,v},\phi_{0,v})={\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}_{v}))q_{v}^{s}\times q_{v}^{d_{v}(s-3/2)}\frac{\zeta_{F,v}(s)}{\zeta_{F,v}(2s)}L(s,\pi_{v};{\operatorname{Ad}}).
Proof.

If vΣ𝑣subscriptΣv\in\Sigma_{\infty}, the integral (3.9) for ϕ0,vsubscriptitalic-ϕ0𝑣\phi_{0,v} is computed in the proof of [26, Lemma 6.4]. (Note that ϕ0,v(vΣ)subscriptitalic-ϕ0𝑣𝑣subscriptΣ\phi_{0,v}(v\in\Sigma_{\infty}) in [26] coincides with πv(1001)ϕ0,vsubscript𝜋𝑣1001subscriptitalic-ϕ0𝑣\pi_{v}(\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix})\phi_{0,v} according to our notation). In the rest of the proof, let vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. Suppose c(πv)=0𝑐subscript𝜋𝑣0c(\pi_{v})=0. Then, πvsubscript𝜋𝑣\pi_{v} is a unitarizable spherical representation of PGL(2,Fv)PGL2subscript𝐹𝑣{\operatorname{PGL}}(2,F_{v}). Thus Vπv𝕂v=ϕ0,vsuperscriptsubscript𝑉subscript𝜋𝑣subscript𝕂𝑣subscriptitalic-ϕ0𝑣V_{\pi_{v}}^{{\mathbb{K}}_{v}}={\mathbb{C}}\phi_{0,v} and Vπv𝕂0(𝔭v)=ϕ0,v+ϕ1,vsuperscriptsubscript𝑉subscript𝜋𝑣subscript𝕂0subscript𝔭𝑣subscriptitalic-ϕ0𝑣subscriptitalic-ϕ1𝑣V_{\pi_{v}}^{{\mathbb{K}}_{0}({\mathfrak{p}}_{v})}={\mathbb{C}}\phi_{0,v}+{\mathbb{C}}\phi_{1,v} with ϕ1,v=πv([ϖv1001])ϕ0,vQ(πv)ϕ0,vsubscriptitalic-ϕ1𝑣subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣𝑄subscript𝜋𝑣subscriptitalic-ϕ0𝑣\phi_{1,v}=\pi_{v}\left(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right]\right)\phi_{0,v}-Q(\pi_{v})\,\phi_{0,v}. By [30, (2.30)], we have the formula (3.11). From the formula of ϕ1,vsubscriptitalic-ϕ1𝑣\phi_{1,v},

Z(s,ϕ1,v,ϕ1,v)=𝑍𝑠subscriptitalic-ϕ1𝑣subscriptitalic-ϕ1𝑣absent\displaystyle Z(s,\phi_{1,v},\phi_{1,v})= Z(s,πv([ϖv1001])ϕ0,v,πv([ϖv1001])ϕ0,v)Q(πv)Z(s,ϕ0,v,πv([ϖv1001])ϕ0,v)𝑍𝑠subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣𝑄subscript𝜋𝑣𝑍𝑠subscriptitalic-ϕ0𝑣subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣\displaystyle Z(s,\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v},\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v})-Q(\pi_{v})Z(s,\phi_{0,v},\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v})
Q(πv)Z(s,πv([ϖv1001])ϕ0,v,ϕ0,v)+Q(πv)2Z(s,ϕ0,v,ϕ0,v).𝑄subscript𝜋𝑣𝑍𝑠subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣𝑄superscriptsubscript𝜋𝑣2𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣\displaystyle-Q(\pi_{v})Z(s,\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v},\phi_{0,v})+Q(\pi_{v})^{2}Z(s,\phi_{0,v},\phi_{0,v}).

As for the first term, by (3.10),

Z(s,πv([ϖv1001])ϕ0,v,πv([ϖv1001])ϕ0,v)𝑍𝑠subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣\displaystyle Z(s,\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v},\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v})
=\displaystyle= Fv×𝐊0(𝔭v)ϕ0,v([t001]k[ϖv1001])ϕ0,v([t001]k[ϖv1001])¯|t|vs1d×t𝑑ksubscriptsuperscriptsubscript𝐹𝑣subscriptsubscript𝐊0subscript𝔭𝑣subscriptitalic-ϕ0𝑣delimited-[]𝑡001𝑘delimited-[]superscriptsubscriptitalic-ϖ𝑣1001¯subscriptitalic-ϕ0𝑣delimited-[]𝑡001𝑘delimited-[]superscriptsubscriptitalic-ϖ𝑣1001superscriptsubscript𝑡𝑣𝑠1superscript𝑑𝑡differential-d𝑘\displaystyle\textstyle{\int}_{F_{v}^{\times}}\textstyle{\int}_{{\mathbf{K}}_{0}({\mathfrak{p}}_{v})}\phi_{0,v}([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}]k[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}])\overline{\phi_{0,v}([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}]k[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}])}|t|_{v}^{s-1}d^{\times}tdk
+ξ𝔬v/𝔭vFv×𝐊0(𝔭v)ϕ0,v([t001][1ξ01]w0k[ϖv1001])ϕ0,v([t001][1ξ01]w0k[ϖv1001])¯|t|vs1d×t𝑑k.subscript𝜉subscript𝔬𝑣subscript𝔭𝑣subscriptsuperscriptsubscript𝐹𝑣subscriptsubscript𝐊0subscript𝔭𝑣subscriptitalic-ϕ0𝑣delimited-[]𝑡001delimited-[]1𝜉01subscript𝑤0𝑘delimited-[]superscriptsubscriptitalic-ϖ𝑣1001¯subscriptitalic-ϕ0𝑣delimited-[]𝑡001delimited-[]1𝜉01subscript𝑤0𝑘delimited-[]superscriptsubscriptitalic-ϖ𝑣1001superscriptsubscript𝑡𝑣𝑠1superscript𝑑𝑡differential-d𝑘\displaystyle+\sum_{\xi\in{\mathfrak{o}}_{v}/{\mathfrak{p}}_{v}}\textstyle{\int}_{F_{v}^{\times}}\textstyle{\int}_{{\mathbf{K}}_{0}({\mathfrak{p}}_{v})}\phi_{0,v}([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}][\begin{smallmatrix}1&\xi\\ 0&1\end{smallmatrix}]w_{0}k[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}])\overline{\phi_{0,v}([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}][\begin{smallmatrix}1&\xi\\ 0&1\end{smallmatrix}]w_{0}k[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}])}|t|_{v}^{s-1}d^{\times}tdk.

Since Ad([ϖv001])𝕂0(𝔭v)𝕂vAddelimited-[]subscriptitalic-ϖ𝑣001subscript𝕂0subscript𝔭𝑣subscript𝕂𝑣{\operatorname{Ad}}(\left[\begin{smallmatrix}\varpi_{v}&0\\ 0&1\end{smallmatrix}\right]){\mathbb{K}}_{0}({\mathfrak{p}}_{v})\subset{\mathbb{K}}_{v}, the first integral of the right-hand side is seen to be equal to vol(𝐊0(𝔭v))qv1sZ(s,ϕ0,v,ϕ0,v)volsubscript𝐊0subscript𝔭𝑣superscriptsubscript𝑞𝑣1𝑠𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣{\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}_{v}))q_{v}^{1-s}Z(s,\phi_{0,v},\phi_{0,v}). Similarly, by the equation ϕ0,v([t001][1ξ01]g)=ψv(tξ)ϕ0,v([t001]g)subscriptitalic-ϕ0𝑣delimited-[]𝑡001delimited-[]1𝜉01𝑔subscript𝜓𝑣𝑡𝜉subscriptitalic-ϕ0𝑣delimited-[]𝑡001𝑔\phi_{0,v}([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}][\begin{smallmatrix}1&\xi\\ 0&1\end{smallmatrix}]g)=\psi_{v}(t\xi)\phi_{0,v}([\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}]g) and by a simple variable change, each of ξ𝜉\xi-terms is computed to be vol(𝐊0(𝔭v))qvs1Z(s,ϕ0,v,ϕ0,v).volsubscript𝐊0subscript𝔭𝑣superscriptsubscript𝑞𝑣𝑠1𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣{\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}_{v}))q_{v}^{s-1}Z(s,\phi_{0,v},\phi_{0,v}). Thus, we have

Z(s,πv([ϖv1001])ϕ0,v,πv([ϖv1001])ϕ0,v)=(1+qv)1(qv1s+qvs)Z(s,ϕ0,v,ϕ0,v).𝑍𝑠subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣superscript1subscript𝑞𝑣1superscriptsubscript𝑞𝑣1𝑠superscriptsubscript𝑞𝑣𝑠𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣\displaystyle Z(s,\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v},\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v})=(1+q_{v})^{-1}(q_{v}^{1-s}+q_{v}^{s})Z(s,\phi_{0,v},\phi_{0,v}).

Similarly, by the relation ϕ([t001])=ϕ([t001])¯italic-ϕdelimited-[]𝑡001¯italic-ϕdelimited-[]𝑡001\phi(\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right])=\overline{\phi(\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right])} which follows from the unitarity of πvsubscript𝜋𝑣\pi_{v} and by (3.2), the integral Z(s,πv([ϖv1001])ϕ0,v,ϕ0,v)𝑍𝑠subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣Z(s,\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v},\phi_{0,v}) is computed as

vol(𝐊0(𝔭v))(qv1s+qv)×qv1/2(αv+αv1)qvdv(s3/2)L(s,πv;Ad).volsubscript𝐊0subscript𝔭𝑣superscriptsubscript𝑞𝑣1𝑠subscript𝑞𝑣superscriptsubscript𝑞𝑣12subscript𝛼𝑣superscriptsubscript𝛼𝑣1superscriptsubscript𝑞𝑣subscript𝑑𝑣𝑠32𝐿𝑠subscript𝜋𝑣Ad{\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}_{v}))(q_{v}^{1-s}+q_{v})\times q_{v}^{-1/2}(\alpha_{v}+\alpha_{v}^{-1})q_{v}^{d_{v}(s-3/2)}L(s,\pi_{v};{\operatorname{Ad}}).

Hence, we obtain

Z(s,πv(ϖv1001)ϕ0,v,ϕ0,v)=(1+qv)1qv1/2(αv+αv1)Z(s,ϕ0,v,ϕ0,v).𝑍𝑠subscript𝜋𝑣superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣superscript1subscript𝑞𝑣1superscriptsubscript𝑞𝑣12subscript𝛼𝑣superscriptsubscript𝛼𝑣1𝑍𝑠subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣Z(s,\pi_{v}(\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix})\phi_{0,v},\phi_{0,v})=(1+q_{v})^{-1}q_{v}^{1/2}(\alpha_{v}+\alpha_{v}^{-1})Z(s,\phi_{0,v},\phi_{0,v}).

In a similar fashion, the equality Z(s,ϕ0,v,πv([ϖv1001])ϕ0,v)=Z(s,πv([ϖv1001])ϕ0,v,ϕ0,v)𝑍𝑠subscriptitalic-ϕ0𝑣subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣𝑍𝑠subscript𝜋𝑣delimited-[]superscriptsubscriptitalic-ϖ𝑣1001subscriptitalic-ϕ0𝑣subscriptitalic-ϕ0𝑣Z(s,\phi_{0,v},\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v})=Z(s,\pi_{v}(\left[\begin{smallmatrix}\varpi_{v}^{-1}&0\\ 0&1\end{smallmatrix}\right])\phi_{0,v},\phi_{0,v}) holds. All in all, we have the formula (3.12).

For the last formula (i.e., c(πv)=1𝑐subscript𝜋𝑣1c(\pi_{v})=1), we refer to the proof of [30, Lemma 2.14]. ∎

By using Lemma 3.3 with [30, Lemmas 2.4 and 2.14] and [26, Lemma 6.4] to compute the summands of (3.7), we obtain

E(z)(l,𝔫;π)=DFz1/22N(𝔣π)1/2z/2{vS(𝔫𝔣π1)(1+Q(Iv(||vz/2))Q(πv)21Q(πv)2)}ζF(z+12)L(z+12,π;Ad)L(1,π;Ad).\displaystyle\mathbb{P}_{E^{*}(z)}(l,{\mathfrak{n}};\pi)=\frac{D_{F}^{z-1/2}}{2{\operatorname{N}}({\mathfrak{f}}_{\pi})^{1/2-z/2}}\{\prod_{v\in S({\mathfrak{n}}{\mathfrak{f}}_{\pi}^{-1})}\biggl{(}1+\frac{Q(I_{v}(|\,|_{v}^{z/2}))-Q(\pi_{v})^{2}}{1-Q(\pi_{v})^{2}}\biggr{)}\}\,\frac{\zeta_{F}\left(\tfrac{z+1}{2}\right)L\left(\tfrac{z+1}{2},\pi;{\operatorname{Ad}}\right)}{L(1,\pi;{\operatorname{Ad}})}.

4. An overview of the geometric side

4.1. Basic assumption

From this section on, until otherwise stated, we let 𝔫𝔫{\mathfrak{n}} be a non-zero ideal of 𝔬𝔬\mathfrak{o}, l=(lv)vΣ𝑙subscriptsubscript𝑙𝑣𝑣subscriptΣl=(l_{v})_{v\in\Sigma_{\infty}} an element of (2)Σsuperscript2subscriptΣ(2{\mathbb{N}})^{\Sigma_{\infty}}, and S𝑆S a finite set of ΣfinsubscriptΣfin\Sigma_{\rm fin}. We keep the following assumptions on (𝔫,S,l)𝔫𝑆𝑙({\mathfrak{n}},S,l):

  • (i)

    The ideal 𝔫𝔬𝔫𝔬{\mathfrak{n}}\subset\mathfrak{o} is square-free.

  • (ii)

    The two sets S𝑆S, S(𝔫)𝑆𝔫S({\mathfrak{n}}) are mutually disjoint.

  • (iii)

    The weight l=(lv)vΣ𝑙subscriptsubscript𝑙𝑣𝑣subscriptΣl=(l_{v})_{v\in\Sigma_{\infty}} is large in the sense that l¯:=minvΣlv4assign¯𝑙subscript𝑣subscriptΣsubscript𝑙𝑣4{\underline{l}}:=\min_{v\in\Sigma_{\infty}}l_{v}\geqslant 4.

Let Σdyadic={vΣfin||2|v<1}subscriptΣdyadicconditional-set𝑣subscriptΣfinsubscript2𝑣1\Sigma_{\rm{dyadic}}=\{v\in\Sigma_{\rm fin}|\,|2|_{v}<1\,\} be the set of all the dyadic places in ΣfinsubscriptΣfin\Sigma_{\rm fin}. After § 7.3, we further suppose

  • (iv)

    SS(𝔫)𝑆𝑆𝔫S\cup S({\mathfrak{n}}) is disjoint from ΣdyadicsubscriptΣdyadic\Sigma_{\rm dyadic}.

  • (v)

    The place 222 of {\mathbb{Q}} splits completely in F/𝐹F/{\mathbb{Q}}.

In this paper, l𝑙l and S𝑆S are fixed. Until §8, the ideal 𝔫𝔫{\mathfrak{n}} is also fixed. To simplify notation, Φl(𝔫|𝐬,g)superscriptΦ𝑙conditional𝔫𝐬𝑔\Phi^{l}({\mathfrak{n}}|{\mathbf{s}},g) and 𝚽l(𝔫|𝐬,g,h)superscript𝚽𝑙conditional𝔫𝐬𝑔{\bf\Phi}^{l}({\mathfrak{n}}|{\mathbf{s}},g,h) are abbreviated to Φ(𝐬;g)Φ𝐬𝑔\Phi({\mathbf{s}};g) and 𝚽(𝐬;g,h)𝚽𝐬𝑔{\bf\Phi}({\mathbf{s}};g,h), respectively.

4.2. An overview

From the classification of conjugacy classes of ZF\GF\subscript𝑍𝐹subscript𝐺𝐹Z_{F}\backslash G_{F}, we have

(4.1) 𝚽(𝐬;g,g)=Jid(𝐬;g)+Jell(𝐬;g)+Jhyp(𝐬;g)+Junip(𝐬;g),𝚽𝐬𝑔𝑔subscript𝐽id𝐬𝑔subscript𝐽ell𝐬𝑔subscript𝐽hyp𝐬𝑔subscript𝐽unip𝐬𝑔\displaystyle{\bf\Phi}({\mathbf{s}};g,g)=J_{\rm{id}}({\mathbf{s}};g)+J_{\rm{ell}}({\mathbf{s}};g)+J_{\rm{hyp}}({\mathbf{s}};g)+J_{\rm{unip}}({\mathbf{s}};g),

where the four terms on the right-hand side are defined as follows ([5], [12]).

(4.2) Jid(𝐬;g)subscript𝐽id𝐬𝑔\displaystyle J_{\rm{id}}({\mathbf{s}};g) =Φ(𝐬;12)(identity term),absentΦ𝐬subscript12identity term\displaystyle={\Phi}({\mathbf{s}};1_{2})\quad(\text{identity term}),
(4.3) Junip(𝐬;g)subscript𝐽unip𝐬𝑔\displaystyle J_{\rm{unip}}({\mathbf{s}};g) =ξZFNF\GFΦ(𝐬;g1ξ1[1101]ξg)(the unipotent term),absentsubscript𝜉\subscript𝑍𝐹subscript𝑁𝐹subscript𝐺𝐹Φ𝐬superscript𝑔1superscript𝜉1delimited-[]1101𝜉𝑔the unipotent term\displaystyle=\sum_{\xi\in Z_{F}N_{F}\backslash G_{F}}{\Phi}({\mathbf{s}};g^{-1}\xi^{-1}\left[\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right]\xi g)\quad(\text{the unipotent term}),
(4.4) Jhyp(𝐬;g)subscript𝐽hyp𝐬𝑔\displaystyle J_{\rm{hyp}}({\mathbf{s}};g) =12ξHF\GFaF×{1}Φ(𝐬;g1ξ1[a001]ξg)(the F-hyperbolic term),absent12subscript𝜉\subscript𝐻𝐹subscript𝐺𝐹subscript𝑎superscript𝐹1Φ𝐬superscript𝑔1superscript𝜉1delimited-[]𝑎001𝜉𝑔the F-hyperbolic term\displaystyle=\frac{1}{2}\sum_{\xi\in H_{F}\backslash G_{F}}\sum_{a\in F^{\times}-\{1\}}{\Phi}({\mathbf{s}};g^{-1}\xi^{-1}\left[\begin{smallmatrix}a&0\\ 0&1\end{smallmatrix}\right]\xi g)\quad(\text{the $F$-hyperbolic term}),
(4.5) Jell(𝐬;g)subscript𝐽ell𝐬𝑔\displaystyle J_{\rm{ell}}({\mathbf{s}};g) =12E𝔈ξTE\GFγZF\(TEZF)Φ(𝐬;g1ξ1γξg)(the F-elliptic term),absent12subscript𝐸𝔈subscript𝜉\subscript𝑇𝐸subscript𝐺𝐹subscript𝛾\subscript𝑍𝐹subscript𝑇𝐸subscript𝑍𝐹Φ𝐬superscript𝑔1superscript𝜉1𝛾𝜉𝑔the F-elliptic term\displaystyle=\frac{1}{2}\sum_{E\in{\mathfrak{E}}}\sum_{\xi\in T_{E}\backslash G_{F}}\sum_{\gamma\in Z_{F}\backslash(T_{E}-Z_{F})}{\Phi}({\mathbf{s}};g^{-1}\xi^{-1}\gamma\xi g)\quad(\text{the $F$-elliptic term}),

where 𝔈𝔈{\mathfrak{E}} is the set of all quadratic division F𝐹F-subalgebra EM2(F)𝐸subscriptM2𝐹E\subset{\operatorname{M}}_{2}(F) and TE=E×subscript𝑇𝐸superscript𝐸T_{E}=E^{\times} viewed as an F𝐹F-elliptic torus of GL(2)GL2{\operatorname{GL}}(2). Then, from Lemma 2.5 and Proposition 3.2, for each type of conjugacy classes {id,unip,hyp,ell}iduniphypell\natural\in\{\rm id,unip,hyp,ell\}, we know that the integral

(4.6) 𝕁(𝐬,β)=Z𝔸GF\G𝔸β(g)J(𝐬;g)𝑑gsubscript𝕁𝐬𝛽subscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸superscriptsubscript𝛽𝑔subscript𝐽𝐬𝑔differential-d𝑔\displaystyle{\mathbb{J}}_{\natural}({\mathbf{s}},\beta)=\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}{{\mathcal{E}}}_{\beta}^{*}(g)\,J_{\natural}({\mathbf{s}};g)\,{{d}}g

is absolutely convergent for β1𝛽subscript1\beta\in{\mathcal{B}}_{1}. In the succeeding three sections, by computing this integral quite explicitly, we shall show the following:

Theorem 4.1.

For any 𝐬𝔛S𝐬subscript𝔛𝑆{\mathbf{s}}\in{\mathfrak{X}}_{S} lying on some domain minvSRe(sv)>σ0subscript𝑣𝑆Resubscript𝑠𝑣subscript𝜎0\min_{v\in S}\operatorname{Re}(s_{v})>\sigma_{0}, there exists a meromorphic function J^(𝐬;z)subscript^𝐽𝐬𝑧\hat{J}_{\natural}({\mathbf{s}};z) in z𝑧z on a vertical strip Re(z)(σ1,σ2)Re𝑧subscript𝜎1subscript𝜎2\operatorname{Re}(z)\in(\sigma_{1},\sigma_{2}) containing 0<Re(z)<10Re𝑧10<\operatorname{Re}(z)<1, which is holomorphic away from z=±1,0𝑧plus-or-minus10z=\pm 1,0, is vertically of moderate growth, and satisfies

(4.7) 𝕁(𝐬,β)=LσJ^(𝐬,z)β(z)𝑑z,β1,σ(σ1,σ2).formulae-sequencesubscript𝕁𝐬𝛽subscriptsubscript𝐿𝜎subscript^𝐽𝐬𝑧𝛽𝑧differential-d𝑧formulae-sequence𝛽subscript1𝜎subscript𝜎1subscript𝜎2\displaystyle{\mathbb{J}}_{\natural}({\mathbf{s}},\beta)=\int_{L_{\sigma}}\hat{J}_{\natural}({\mathbf{s}},z)\,\beta(z)\,{{d}}z,\quad\beta\in{\mathcal{B}}_{1},\quad\sigma\in(\sigma_{1},\sigma_{2}).

Fix 𝐬𝔛S𝐬subscript𝔛𝑆{\mathbf{s}}\in{\mathfrak{X}}_{S} with minvSRe(sv)>σ0subscript𝑣𝑆Resubscript𝑠𝑣subscript𝜎0\min_{v\in S}\operatorname{Re}(s_{v})>\sigma_{0}. From (3.8) and (4.1),

LσI^cusp(𝐬,z)β(z)𝑑z=Lσ{J^id(𝐬,z)+J^unip(𝐬,z)+J^hyp(𝐬,z)+J^ell(𝐬,z)}β(z)𝑑zsubscriptsubscript𝐿𝜎subscript^𝐼cusp𝐬𝑧𝛽𝑧differential-d𝑧subscriptsubscript𝐿𝜎subscript^𝐽id𝐬𝑧subscript^𝐽unip𝐬𝑧subscript^𝐽hyp𝐬𝑧subscript^𝐽ell𝐬𝑧𝛽𝑧differential-d𝑧\int_{L_{\sigma}}\hat{I}_{\rm{cusp}}({\mathbf{s}},z)\beta(z)\,{{d}}z=\int_{L_{\sigma}}\{\hat{J}_{\rm id}({\mathbf{s}},z)+\hat{J}_{\rm unip}({\mathbf{s}},z)+\hat{J}_{\rm hyp}({\mathbf{s}},z)+\hat{J}_{\rm ell}({\mathbf{s}},z)\}\beta(z){{d}}z

for all β1𝛽subscript1\beta\in{\mathcal{B}}_{1} and σ(σ1,σ2)𝜎subscript𝜎1subscript𝜎2\sigma\in(\sigma_{1},\sigma_{2}). By applying Lemma 4.2 below, noting (3.8) and (4.1), we obtain the trace formula in the form I^cusp(𝐬,z)=J^(𝐬,z)subscript^𝐼cusp𝐬𝑧subscriptsubscript^𝐽𝐬𝑧\hat{I}_{\rm cusp}({\mathbf{s}},z)=\sum_{\natural}\hat{J}_{\natural}({\mathbf{s}},z) as a meromorphic function on Re(z)(σ1,σ2)Re𝑧subscript𝜎1subscript𝜎2\operatorname{Re}(z)\in(\sigma_{1},\sigma_{2}).

Lemma 4.2.

Let F(z)𝐹𝑧F(z) be a meromorphic function on a strip Re(z)(σ1,σ2)Re𝑧subscript𝜎1subscript𝜎2\operatorname{Re}(z)\in(\sigma_{1},\sigma_{2}), which is vertically of moderate growth and is holomorphic away from possible simple poles at z=0,±1𝑧0plus-or-minus1z=0,\pm 1. Assume that there exists σ(σ1,σ2)𝜎subscript𝜎1subscript𝜎2\sigma\in(\sigma_{1},\sigma_{2}), σ0,±1𝜎0plus-or-minus1\sigma\not=0,\pm 1 such that LσF(z)β(z)𝑑z=0subscriptsubscript𝐿𝜎𝐹𝑧𝛽𝑧differential-d𝑧0\int_{L_{\sigma}}F(z)\beta(z)dz=0 for all β1𝛽subscript1\beta\in{\mathcal{B}}_{1}. Then we have F(z)=0𝐹𝑧0F(z)=0 identically on the strip Re(z)(σ1,σ2)Re𝑧subscript𝜎1subscript𝜎2\operatorname{Re}(z)\in(\sigma_{1},\sigma_{2}).

Proof.

Set F1(z)=z(z21)2F(z)subscript𝐹1𝑧𝑧superscriptsuperscript𝑧212𝐹𝑧F_{1}(z)=z(z^{2}-1)^{2}F(z). The function A(t)=F1(σ+it)eσ2+2iσtt2/2𝐴𝑡subscript𝐹1𝜎𝑖𝑡superscript𝑒superscript𝜎22𝑖𝜎𝑡superscript𝑡22A(t)=F_{1}(\sigma+it)e^{\sigma^{2}+2i\sigma t-t^{2}/2} belongs to L2()superscript𝐿2L^{2}({\mathbb{R}}) due to the assumption that F(z)𝐹𝑧F(z) is vertically of moderate growth and to the presence of the factor et2/2superscript𝑒superscript𝑡22e^{-t^{2}/2}. For P(z)[z]𝑃𝑧delimited-[]𝑧P(z)\in{\mathbb{C}}[z], by assumption, we see

A(t)P(σ+it)et2/2𝑑t=LσF(z)×z(z21)2ez2P(z)𝑑z=0.superscriptsubscript𝐴𝑡𝑃𝜎𝑖𝑡superscript𝑒superscript𝑡22differential-d𝑡subscriptsubscript𝐿𝜎𝐹𝑧𝑧superscriptsuperscript𝑧212superscript𝑒superscript𝑧2𝑃𝑧differential-d𝑧0\textstyle{\int}_{-\infty}^{\infty}A(t)P(\sigma+it)e^{-t^{2}/2}dt=\textstyle{\int}_{L_{\sigma}}F(z)\times z(z^{2}-1)^{2}e^{z^{2}}P(z)\,dz=0.

Thus A=0𝐴0A=0 in L2()superscript𝐿2L^{2}({\mathbb{R}}). Since A𝐴A is continuous, we have the pointwise equality F1(σ+it)=0subscript𝐹1𝜎𝑖𝑡0F_{1}(\sigma+it)=0 for all t𝑡t\in{\mathbb{R}}. By the holomorphicity, F1(z)=0subscript𝐹1𝑧0F_{1}(z)=0 identically. ∎

5. The singular terms

5.1. The identity term

We shall see 𝕁id(𝐬,β)=0subscript𝕁id𝐬𝛽0{\mathbb{J}}_{\rm{id}}({\mathbf{s}},\beta)=0. Let σ>1𝜎1\sigma>1. Since γBF\GFy(γg)σ+12subscript𝛾\subscript𝐵𝐹subscript𝐺𝐹𝑦superscript𝛾𝑔𝜎12\sum_{\gamma\in B_{F}\backslash G_{F}}y(\gamma g)^{\frac{\sigma+1}{2}} is convergent,

Z𝔸GF\G𝔸β(g)𝑑gsubscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸superscriptsubscript𝛽𝑔differential-d𝑔\displaystyle\textstyle{\int}_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}{\mathcal{E}}_{\beta}^{*}(g)\,{{d}}g =Z𝔸GF\G𝔸γBF\GF(Lσy(γg)z+12ΛF(z+1)β(z)𝑑z)dgabsentsubscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸subscript𝛾\subscript𝐵𝐹subscript𝐺𝐹subscriptsubscript𝐿𝜎𝑦superscript𝛾𝑔𝑧12subscriptΛ𝐹𝑧1𝛽𝑧differential-d𝑧𝑑𝑔\displaystyle=\textstyle{\int}_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}\sum_{\gamma\in B_{F}\backslash G_{F}}\left(\textstyle{\int}_{L_{\sigma}}y(\gamma g)^{\frac{z+1}{2}}\,\Lambda_{F}(z+1)\,\beta(z)\,{{d}}z\right)\,{{d}}g
=Z𝔸BF\G𝔸(Lσy(g)z+12ΛF(z+1)β(z)𝑑z)𝑑g=S0+S,absentsubscript\subscript𝑍𝔸subscript𝐵𝐹subscript𝐺𝔸subscriptsubscript𝐿𝜎𝑦superscript𝑔𝑧12subscriptΛ𝐹𝑧1𝛽𝑧differential-d𝑧differential-d𝑔subscript𝑆0subscript𝑆\displaystyle=\textstyle{\int}_{Z_{\mathbb{A}}B_{F}\backslash G_{\mathbb{A}}}\left(\int_{L_{\sigma}}y(g)^{\frac{z+1}{2}}\,\Lambda_{F}(z+1)\,\beta(z)\,{{d}}z\right)\,{{d}}g=S_{0}+S_{\infty},

where S0subscript𝑆0S_{0} and Ssubscript𝑆S_{\infty} denote the integrals over the subdomains y(g)<1𝑦𝑔1y(g)<1 and y(g)1𝑦𝑔1y(g)\geqslant 1, respectively. By writing gZ𝔸\G𝔸𝑔\subscript𝑍𝔸subscript𝐺𝔸g\in Z_{\mathbb{A}}\backslash G_{\mathbb{A}} as [1x01][t¯u001]kdelimited-[]1𝑥01delimited-[]¯𝑡𝑢001𝑘\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}\underline{t}\,u&0\\ 0&1\end{smallmatrix}\right]k with x𝔸,t>0,u𝔸1,k𝕂formulae-sequence𝑥𝔸formulae-sequence𝑡0formulae-sequence𝑢superscript𝔸1𝑘𝕂x\in{\mathbb{A}},\,t>0,\,u\in{\mathbb{A}}^{1},\,k\in{\mathbb{K}}, and by changing the order of integrals, we have

S0subscript𝑆0\displaystyle S_{0} =Lσ(F\𝔸𝑑xF×\𝔸1d1u01tz12d×t)ΛF(z+1)β(z)𝑑z=vol(F×\𝔸1)Lσ2ΛF(z+1)β(z)z1𝑑z.absentsubscriptsubscript𝐿𝜎subscript\𝐹𝔸differential-d𝑥subscript\superscript𝐹superscript𝔸1superscript𝑑1𝑢superscriptsubscript01superscript𝑡𝑧12superscript𝑑𝑡subscriptΛ𝐹𝑧1𝛽𝑧differential-d𝑧vol\superscript𝐹superscript𝔸1subscriptsubscript𝐿𝜎2subscriptΛ𝐹𝑧1𝛽𝑧𝑧1differential-d𝑧\displaystyle=\textstyle{\int}_{L_{\sigma}}\left(\int_{F\backslash{\mathbb{A}}}{{d}}x\textstyle{\int}_{F^{\times}\backslash{\mathbb{A}}^{1}}{{d}}^{1}u\textstyle{\int}_{0}^{1}t^{\frac{z-1}{2}}{{d}}^{\times}t\right)\Lambda_{F}(z+1)\beta(z)\,{{d}}z={\operatorname{vol}}(F^{\times}\backslash{\mathbb{A}}^{1})\,\textstyle{\int}_{L_{\sigma}}\tfrac{2\Lambda_{F}(z+1)\beta(z)}{z-1}\,{{d}}z.

As for Ssubscript𝑆S_{\infty}, shifting the contour Lσsubscript𝐿𝜎L_{\sigma} to Lσsubscript𝐿𝜎L_{-\sigma} and then in the same way as above, we have

Ssubscript𝑆\displaystyle S_{\infty} =Z𝔸BF\G𝔸y(g)>1(Lσy(g)z12ΛF(z+1)β(z)𝑑z)𝑑g=vol(F×\𝔸1)Lσ2ΛF(z+1)β(z)z1𝑑z.absentsubscript\subscript𝑍𝔸subscript𝐵𝐹subscript𝐺𝔸𝑦𝑔1subscriptsubscript𝐿𝜎𝑦superscript𝑔𝑧12subscriptΛ𝐹𝑧1𝛽𝑧differential-d𝑧differential-d𝑔vol\superscript𝐹superscript𝔸1subscriptsubscript𝐿𝜎2subscriptΛ𝐹𝑧1𝛽𝑧𝑧1differential-d𝑧\displaystyle=\textstyle{\int}_{\begin{subarray}{c}Z_{\mathbb{A}}B_{F}\backslash G_{\mathbb{A}}\\ y(g)>1\end{subarray}}\left(\textstyle{\int}_{L_{-\sigma}}y(g)^{\frac{z-1}{2}}\,\Lambda_{F}(z+1)\beta(z)\,{{d}}z\right)\,{{d}}g={\operatorname{vol}}(F^{\times}\backslash{\mathbb{A}}^{1})\textstyle{\int}_{L_{-\sigma}}\tfrac{-2\Lambda_{F}(z+1)\beta(z)}{z-1}\,{{d}}z.

Thus, by the residue theorem, we see that vol(F×\𝔸1)1(S0+S){\operatorname{vol}}(F^{\times}\backslash{\mathbb{A}}^{1})^{-1}(S_{0}+S_{\infty}) equals Resz=12ΛF(z+1)β(z)z1=2DF1/2ζF(2)β(1)=0subscriptRes𝑧12subscriptΛ𝐹𝑧1𝛽𝑧𝑧12superscriptsubscript𝐷𝐹12subscript𝜁𝐹2𝛽10{\rm{Res}}_{z=1}\tfrac{2\Lambda_{F}(z+1)\beta(z)}{z-1}=2D_{F}^{1/2}\zeta_{F}(2)\beta(1)=0. By 𝕁id(𝐬,β)=Φ(𝐬;12)(S0+S)subscript𝕁id𝐬𝛽Φ𝐬subscript12subscript𝑆0subscript𝑆{\mathbb{J}}_{\rm{id}}({\mathbf{s}},\beta)=\Phi({\mathbf{s}};1_{2})(S_{0}+S_{\infty}), we obtain 𝕁id(𝐬,β)=0subscript𝕁id𝐬𝛽0{\mathbb{J}}_{\rm{id}}({\mathbf{s}},\beta)=0. If we set J^id(𝐬,z)=0subscript^𝐽id𝐬𝑧0\hat{J}_{\rm{id}}({\mathbf{s}},z)=0, then this shows that Theorem 4.1 is valid for the identity term.

5.2. The unipotent term

By noting vol(NF\N𝔸)=1vol\subscript𝑁𝐹subscript𝑁𝔸1{\operatorname{vol}}(N_{F}\backslash N_{\mathbb{A}})=1, from (4.3) and (4.6), we have

𝕁unip(𝐬,β)=Z𝔸N𝔸\G𝔸Φ(𝐬;g1[1101]g)β,(g)𝑑g.subscript𝕁unip𝐬𝛽subscript\subscript𝑍𝔸subscript𝑁𝔸subscript𝐺𝔸Φ𝐬superscript𝑔1delimited-[]1101𝑔superscriptsubscript𝛽𝑔differential-d𝑔\displaystyle{\mathbb{J}}_{\rm unip}({\mathbf{s}},\beta)=\int_{Z_{\mathbb{A}}N_{\mathbb{A}}\backslash G_{\mathbb{A}}}\Phi({\mathbf{s}};g^{-1}[\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}]g){\mathcal{E}}_{\beta,\circ}^{*}(g)dg.

Here β,(g)=NF\N𝔸β([1x01]g)𝑑xsuperscriptsubscript𝛽𝑔subscript\subscript𝑁𝐹subscript𝑁𝔸superscriptsubscript𝛽delimited-[]1𝑥01𝑔differential-d𝑥{\mathcal{E}}_{\beta,\circ}^{*}(g)=\int_{N_{F}\backslash N_{\mathbb{A}}}{\mathcal{E}}_{\beta}^{*}([\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}]g)dx is the constant term which is computed as

β,(g)=Lσβ(z){ΛF(z)y(g)(1+z)/2+ΛF(z)y(g)(1z)/2}𝑑z.superscriptsubscript𝛽𝑔subscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧𝑦superscript𝑔1𝑧2subscriptΛ𝐹𝑧𝑦superscript𝑔1𝑧2differential-d𝑧{\mathcal{E}}_{\beta,\circ}^{*}(g)=\int_{L_{\sigma}}\beta(z)\{\Lambda_{F}(-z)y(g)^{(1+z)/2}+\Lambda_{F}(z)y(g)^{(1-z)/2}\}dz.

By substituting this to the formula of 𝕁unip(𝐬,β)subscript𝕁unip𝐬𝛽{\mathbb{J}}_{\rm{unip}}({\mathbf{s}},\beta) and by exchanging the order of integrals formally, we encounter the integral

(5.1) U(𝐬;w)=Z𝔸N𝔸\G𝔸Φ(𝐬;g1[1101]g)y(g)w𝑑g.𝑈𝐬𝑤subscript\subscript𝑍𝔸subscript𝑁𝔸subscript𝐺𝔸Φ𝐬superscript𝑔1delimited-[]1101𝑔𝑦superscript𝑔𝑤differential-d𝑔\displaystyle U({\mathbf{s}};w)=\int_{Z_{\mathbb{A}}N_{\mathbb{A}}\backslash G_{\mathbb{A}}}\Phi({\mathbf{s}};g^{-1}[\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}]g)y(g)^{w}dg.

In order to analyze this, we consider the local integrals

Uv(w)=ZvNv\GvΦv(g1[1101]g)y(g)w𝑑g=Fv×𝐊vΦv(k1[1t01]k)|t|v1wd×t𝑑ksubscript𝑈𝑣𝑤subscript\subscript𝑍𝑣subscript𝑁𝑣subscript𝐺𝑣subscriptΦ𝑣superscript𝑔1delimited-[]1101𝑔𝑦superscript𝑔𝑤differential-d𝑔subscriptsuperscriptsubscript𝐹𝑣subscriptsubscript𝐊𝑣subscriptΦ𝑣superscript𝑘1delimited-[]1𝑡01𝑘superscriptsubscript𝑡𝑣1𝑤superscript𝑑𝑡differential-d𝑘U_{v}(w)=\int_{Z_{v}N_{v}\backslash G_{v}}\Phi_{v}(g^{-1}[\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}]g)y(g)^{w}dg=\int_{F_{v}^{\times}}\int_{{\mathbf{K}}_{v}}\Phi_{v}(k^{-1}[\begin{smallmatrix}1&t\\ 0&1\end{smallmatrix}]k)|t|_{v}^{1-w}d^{\times}tdk

for any vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}, where Φv(gv)subscriptΦ𝑣subscript𝑔𝑣\Phi_{v}(g_{v}) denotes the v𝑣v-th factor of Φ(𝐬;g)Φ𝐬𝑔\Phi({\mathbf{s}};g).

Lemma 5.1.
  • (i)

    For any vΣ𝑣subscriptΣv\in\Sigma_{\infty} and w𝑤w\in\mathbb{C} such that 1lv<Re(w)<11subscript𝑙𝑣Re𝑤11-l_{v}<\operatorname{Re}(w)<1,

    Uv(w)=Γ(1w)×222wπ1w/2Γ(lv+w1)Γ(w/2)1Γ(lv)1.subscript𝑈𝑣𝑤subscriptΓ1𝑤superscript222𝑤superscript𝜋1𝑤2Γsubscript𝑙𝑣𝑤1Γsuperscript𝑤21Γsuperscriptsubscript𝑙𝑣1U_{v}(w)=\Gamma_{\mathbb{R}}(1-w)\times 2^{2-2w}\pi^{1-w/2}{\Gamma(l_{v}+w-1)}{\Gamma(w/2)^{-1}\Gamma(l_{v})^{-1}}.
  • (ii)

    For any vS𝑣𝑆v\in S and (s,w)2𝑠𝑤superscript2(s,w)\in\mathbb{C}^{2} such that Re(s)>1Re𝑠1\operatorname{Re}(s)>1 and Re(s)<Re(w)<1Re𝑠Re𝑤1-\operatorname{Re}(s)<\operatorname{Re}(w)<1, we have

    Uv(w)=qvdv/2×(qv(s+1)/2)(1qvw1)1(1qvsw)1.subscript𝑈𝑣𝑤superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscript𝑞𝑣𝑠12superscript1superscriptsubscript𝑞𝑣𝑤11superscript1superscriptsubscript𝑞𝑣𝑠𝑤1U_{v}(w)=q_{v}^{-d_{v}/2}\times(-q_{v}^{-(s+1)/2})(1-q_{v}^{w-1})^{-1}(1-q_{v}^{-s-w})^{-1}.
  • (iii)

    For any vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}) and w𝑤w\in\mathbb{C} such that Re(w)<1Re𝑤1\operatorname{Re}(w)<1, we have

    Uv(w)=qvdv/2(1+qvw)(1+qv)1(1qvw1)1.subscript𝑈𝑣𝑤superscriptsubscript𝑞𝑣subscript𝑑𝑣21superscriptsubscript𝑞𝑣𝑤superscript1subscript𝑞𝑣1superscript1superscriptsubscript𝑞𝑣𝑤11U_{v}(w)=q_{v}^{-d_{v}/2}(1+q_{v}^{w})(1+q_{v})^{-1}(1-q_{v}^{w-1})^{-1}.
  • (iv)

    For any vΣfin(SS(𝔫))𝑣subscriptΣfin𝑆𝑆𝔫v\in\Sigma_{{\rm fin}}-(S\cup S({\mathfrak{n}})) and w𝑤w\in\mathbb{C} such that Re(w)<1Re𝑤1\operatorname{Re}(w)<1, we have

    Uv(w)=qvdv/2(1qvw1)1.subscript𝑈𝑣𝑤superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscript1superscriptsubscript𝑞𝑣𝑤11U_{v}(w)={q_{v}^{-d_{v}/2}}({1-q_{v}^{w-1}})^{-1}.
Proof.

(i) By (2.1) and by using [9, 3.194, 3], we obtain

Uv(w)=subscript𝑈𝑣𝑤absent\displaystyle U_{v}(w)= Fv×(1it/2)lv|t|w𝑑t=0(1it/2)lvtw𝑑t+0(1+it/2)lvtw𝑑tsubscriptsuperscriptsubscript𝐹𝑣superscript1𝑖𝑡2subscript𝑙𝑣superscript𝑡𝑤differential-d𝑡superscriptsubscript0superscript1𝑖𝑡2subscript𝑙𝑣superscript𝑡𝑤differential-d𝑡superscriptsubscript0superscript1𝑖𝑡2subscript𝑙𝑣superscript𝑡𝑤differential-d𝑡\displaystyle\textstyle{\int}_{F_{v}^{\times}}\left({1-it/2}\right)^{-l_{v}}|t|^{-w}dt=\textstyle{\int}_{0}^{\infty}\left({1-it/2}\right)^{-l_{v}}t^{-w}dt+\int_{0}^{\infty}\left({1+it/2}\right)^{-l_{v}}t^{-w}dt
=\displaystyle= (i/2)w1B(w+1,lv+w1)+(i/2)w1B(w+1,lv+w1)superscript𝑖2𝑤1𝐵𝑤1subscript𝑙𝑣𝑤1superscript𝑖2𝑤1𝐵𝑤1subscript𝑙𝑣𝑤1\displaystyle(-i/2)^{w-1}B(-w+1,l_{v}+w-1)+(i/2)^{w-1}B(-w+1,l_{v}+w-1)
=\displaystyle= 22wsin(π2w)B(w+1,lv+w1),superscript22𝑤𝜋2𝑤𝐵𝑤1subscript𝑙𝑣𝑤1\displaystyle 2^{2-w}\sin\left(\tfrac{\pi}{2}w\right)B(-w+1,l_{v}+w-1),

where B(x,y)=Γ(x)Γ(y)Γ(x+y)𝐵𝑥𝑦Γ𝑥Γ𝑦Γ𝑥𝑦B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} is the beta function. By using the formulas sin(πw/2)=πΓ(w/2)1Γ(1w/2)1𝜋𝑤2𝜋Γsuperscript𝑤21Γsuperscript1𝑤21\sin(\pi w/2)=\pi\Gamma(w/2)^{-1}\Gamma(1-w/2)^{-1} and Γ(1w)=2wπ1/2Γ(1/2w/2)Γ(1w/2)Γ1𝑤superscript2𝑤superscript𝜋12Γ12𝑤2Γ1𝑤2\Gamma(1-w)=2^{-w}\pi^{-1/2}\Gamma(1/2-w/2)\Gamma(1-w/2), we are done.

(ii) From (2.3), Uv(w)=(qv(s+1)/2qv(s+1)/2)1Fv×max(1,|t|v)(s+1)|t|v1wd×tU_{v}(w)=({q_{v}^{-(s+1)/2}-q_{v}^{(s+1)/2}})^{-1}\textstyle{\int}_{F_{v}^{\times}}\max(1,|t|_{v})^{-(s+1)}|t|_{v}^{1-w}d^{\times}t. We compute the integral by dividing the integral domain to |t|v>1subscript𝑡𝑣1|t|_{v}>1 and |t|v1subscript𝑡𝑣1|t|_{v}\leqslant 1 to obtain the desired formula.

(iii) Let tFv×𝑡superscriptsubscript𝐹𝑣t\in F_{v}^{\times} and k=[k11k12k21k22]𝐊v𝑘delimited-[]subscript𝑘11subscript𝑘12subscript𝑘21subscript𝑘22subscript𝐊𝑣k=[\begin{smallmatrix}k_{11}&k_{12}\\ k_{21}&k_{22}\end{smallmatrix}]\in{\mathbf{K}}_{v}. Then, k1[1t01]kZv𝐊0(𝔭v)superscript𝑘1delimited-[]1𝑡01𝑘subscript𝑍𝑣subscript𝐊0subscript𝔭𝑣k^{-1}[\begin{smallmatrix}1&t\\ 0&1\end{smallmatrix}]k\in Z_{v}{\mathbf{K}}_{0}({\mathfrak{p}}_{v}) if and only if t𝔬v×,k21𝔭vformulae-sequence𝑡superscriptsubscript𝔬𝑣subscript𝑘21subscript𝔭𝑣t\in{\mathfrak{o}}_{v}^{\times},k_{21}\in{\mathfrak{p}}_{v} or t𝔭v{0},k21𝔬vformulae-sequence𝑡subscript𝔭𝑣0subscript𝑘21subscript𝔬𝑣t\in{\mathfrak{p}}_{v}-\{0\},k_{21}\in{\mathfrak{o}}_{v}. Thus, Uv(w)subscript𝑈𝑣𝑤U_{v}(w) is computed as the sum of the integral t𝔬v×𝐊0(𝔭v)𝑑k|t|v1wd×t=qvdv/2vol(𝐊0(𝔭v))subscript𝑡superscriptsubscript𝔬𝑣subscriptsubscript𝐊0subscript𝔭𝑣differential-d𝑘superscriptsubscript𝑡𝑣1𝑤superscript𝑑𝑡superscriptsubscript𝑞𝑣subscript𝑑𝑣2volsubscript𝐊0subscript𝔭𝑣\int_{t\in{\mathfrak{o}}_{v}^{\times}}\int_{{\mathbf{K}}_{0}({\mathfrak{p}}_{v})}{{d}}k\,|t|_{v}^{1-w}d^{\times}t=q_{v}^{-d_{v}/2}{\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}_{v})) and the integral t𝔭v{0}𝐊v|t|v1wd×t𝑑k=qvdv/2qvw11qvw1subscript𝑡subscript𝔭𝑣0subscriptsubscript𝐊𝑣superscriptsubscript𝑡𝑣1𝑤superscript𝑑𝑡differential-d𝑘superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscript𝑞𝑣𝑤11superscriptsubscript𝑞𝑣𝑤1\int_{t\in{\mathfrak{p}}_{v}-\{0\}}\int_{{\mathbf{K}}_{v}}|t|_{v}^{1-w}d^{\times}tdk=q_{v}^{-d_{v}/2}\frac{q_{v}^{w-1}}{1-q_{v}^{w-1}}.

(iv) Since Uv(w)=𝔬v{0}|t|v1wd×tsubscript𝑈𝑣𝑤subscriptsubscript𝔬𝑣0superscriptsubscript𝑡𝑣1𝑤superscript𝑑𝑡U_{v}(w)=\int_{{\mathfrak{o}}_{v}-\{0\}}|t|_{v}^{1-w}d^{\times}t, we have the formula immediately. ∎

Suppose that minvSRe(sv)>1subscript𝑣𝑆Resubscript𝑠𝑣1\min_{v\in S}\operatorname{Re}(s_{v})>1, minvSRe(sv)<Re(w)<0subscript𝑣𝑆Resubscript𝑠𝑣Re𝑤0-\min_{v\in S}\operatorname{Re}(s_{v})<\operatorname{Re}(w)<0 and 1l¯<Re(w)<01¯𝑙Re𝑤01-{\underline{l}}<\operatorname{Re}(w)<0 are satisfied. Then from Lemma 5.1 we see that the integral (5.1) converges absolutely and have the formula

U(𝐬;w)=𝑈𝐬𝑤absent\displaystyle U({\mathbf{s}};w)= DF1/2ζF(1w)vΣ222wπ1w/2Γ(lv+w1)Γ(w/2)Γ(lv)vSqv(sv+1)/21qvsvwvS(𝔫)1+qvw1+qv,superscriptsubscript𝐷𝐹12subscript𝜁𝐹1𝑤subscriptproduct𝑣subscriptΣsuperscript222𝑤superscript𝜋1𝑤2Γsubscript𝑙𝑣𝑤1Γ𝑤2Γsubscript𝑙𝑣subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣subscript𝑠𝑣121superscriptsubscript𝑞𝑣subscript𝑠𝑣𝑤subscriptproduct𝑣𝑆𝔫1superscriptsubscript𝑞𝑣𝑤1subscript𝑞𝑣\displaystyle D_{F}^{-1/2}\zeta_{F}(1-w)\prod_{v\in\Sigma_{\infty}}2^{2-2w}\pi^{1-w/2}\frac{\Gamma(l_{v}+w-1)}{\Gamma(w/2)\Gamma(l_{v})}\,\prod_{v\in S}\frac{-q_{v}^{-(s_{v}+1)/2}}{1-q_{v}^{-s_{v}-w}}\prod_{v\in S({\mathfrak{n}})}\frac{1+q_{v}^{w}}{1+q_{v}},

which gives a meromorphic continuation of wU(𝐬;w)maps-to𝑤𝑈𝐬𝑤w\mapsto U({\mathbf{s}};w) to \mathbb{C} for a fixed 𝐬𝐬{\mathbf{s}}. By changing the order of integrals, we obtain

𝕁unip(𝐬,β)=subscript𝕁unip𝐬𝛽absent\displaystyle{\mathbb{J}}_{\rm unip}({\mathbf{s}},\beta)= Z𝔸N𝔸\G𝔸Φl(𝐬|𝔫;g1[1101]g)β,(g)𝑑gsubscript\subscript𝑍𝔸subscript𝑁𝔸subscript𝐺𝔸superscriptΦ𝑙conditional𝐬𝔫superscript𝑔1delimited-[]1101𝑔superscriptsubscript𝛽𝑔differential-d𝑔\displaystyle\int_{Z_{\mathbb{A}}N_{\mathbb{A}}\backslash G_{\mathbb{A}}}\Phi^{l}({\mathbf{s}}|{\mathfrak{n}};g^{-1}[\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}]g){\mathcal{E}}_{\beta,\circ}^{*}(g)dg
=\displaystyle= Lσβ(z)ΛF(z)Ul,S,𝔫(𝐬;(1+z)/2)𝑑z+Lσβ(z)ΛF(z)Ul,S,𝔫(𝐬;(1z)/2)𝑑zsubscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧subscript𝑈𝑙𝑆𝔫𝐬1𝑧2differential-d𝑧subscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧subscript𝑈𝑙𝑆𝔫𝐬1𝑧2differential-d𝑧\displaystyle\int_{L_{-\sigma}}\beta(z)\Lambda_{F}(-z)U_{l,S,{\mathfrak{n}}}({\mathbf{s}};(1+z)/2)dz+\int_{L_{\sigma}}\beta(z)\Lambda_{F}(z)U_{l,S,{\mathfrak{n}}}({\mathbf{s}};(1-z)/2)dz
=\displaystyle= Lσβ(z)J^unip(𝐬,z)𝑑z,subscriptsubscript𝐿𝜎𝛽𝑧subscript^𝐽unip𝐬𝑧differential-d𝑧\displaystyle\int_{L_{\sigma}}\beta(z)\hat{J}_{\rm unip}({\mathbf{s}},z)dz,

where we set

J^unip(𝐬,z)=DFz4ζF(1+z2){J^unip0(𝐬,z)+J^unip0(𝐬,z)}subscript^𝐽unip𝐬𝑧superscriptsubscript𝐷𝐹𝑧4subscript𝜁𝐹1𝑧2superscriptsubscript^𝐽unip0𝐬𝑧superscriptsubscript^𝐽unip0𝐬𝑧\hat{J}_{\rm unip}({\mathbf{s}},z)=D_{F}^{\frac{z}{4}}\,\zeta_{F}\left(\tfrac{1+z}{2}\right)\{\hat{J}_{\rm unip}^{0}({\mathbf{s}},z)+\hat{J}_{\rm unip}^{0}({\mathbf{s}},-z)\}

with

(5.2) J^unip0(𝐬,z)=DFz24ΛF(z)vΣ21zπ3z4Γ(lv+z12)Γ(z+14)Γ(lv)vSqvsv+121qvsvz+12vS(𝔫)1+qvz+121+qv.superscriptsubscript^𝐽unip0𝐬𝑧superscriptsubscript𝐷𝐹𝑧24subscriptΛ𝐹𝑧subscriptproduct𝑣subscriptΣsuperscript21𝑧superscript𝜋3𝑧4Γsubscript𝑙𝑣𝑧12Γ𝑧14Γsubscript𝑙𝑣subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣subscript𝑠𝑣121superscriptsubscript𝑞𝑣subscript𝑠𝑣𝑧12subscriptproduct𝑣𝑆𝔫1superscriptsubscript𝑞𝑣𝑧121subscript𝑞𝑣\displaystyle\hat{J}_{\rm unip}^{0}({\mathbf{s}},z)=D_{F}^{\frac{z-2}{4}}\Lambda_{F}(-z)\prod_{v\in\Sigma_{\infty}}2^{1-z}\pi^{\frac{3-z}{4}}\frac{\Gamma\left(l_{v}+\frac{z-1}{2}\right)}{\Gamma\left(\tfrac{z+1}{4}\right)\Gamma(l_{v})}\,\prod_{v\in S}\frac{-q_{v}^{-\frac{s_{v}+1}{2}}}{1-q_{v}^{-s_{v}-\frac{z+1}{2}}}\prod_{v\in S({\mathfrak{n}})}\frac{1+q_{v}^{\frac{z+1}{2}}}{1+q_{v}}.

Then we obtain Theorem 4.1 for the unipotent term with σ0=σ1=σ2=2l¯1subscript𝜎0subscript𝜎1subscript𝜎22¯𝑙1\sigma_{0}=-\sigma_{1}=\sigma_{2}=2{\underline{l}}-1.

6. The F𝐹F-hyperbolic term

In this section, we study 𝕁hyp(𝐬,β)subscript𝕁hyp𝐬𝛽{\mathbb{J}}_{\rm hyp}({\mathbf{s}},\beta) to show Theorem 4.1 for the F𝐹F-hyperbolic term.

6.1. Spherical functions

First, we recall the explicit formula of 𝕂vsubscript𝕂𝑣{\mathbb{K}}_{v}-invariant spherical functions on Hv\Gv\subscript𝐻𝑣subscript𝐺𝑣H_{v}\backslash G_{v}. Let (w,z)2𝑤𝑧superscript2(w,z)\in{\mathbb{C}}^{2} and vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}. By the Iwasawa decomposition Gv=HvNv𝕂vsubscript𝐺𝑣subscript𝐻𝑣subscript𝑁𝑣subscript𝕂𝑣G_{v}=H_{v}N_{v}{\mathbb{K}}_{v}, we have a well-defined smooth function φv(w,z):Gv:superscriptsubscript𝜑𝑣𝑤𝑧subscript𝐺𝑣\varphi_{v}^{(w,z)}:G_{v}\rightarrow{\mathbb{C}} such that

(6.1) φv(w,z)([t100t2][1x01]k)=|t1/t2|vw{Av(w,z)hv(w,z)(x)+Av(w,z)hv(w,z)(x)},superscriptsubscript𝜑𝑣𝑤𝑧delimited-[]subscript𝑡100subscript𝑡2delimited-[]1𝑥01𝑘superscriptsubscriptsubscript𝑡1subscript𝑡2𝑣𝑤subscript𝐴𝑣𝑤𝑧superscriptsubscript𝑣𝑤𝑧𝑥subscript𝐴𝑣𝑤𝑧superscriptsubscript𝑣𝑤𝑧𝑥\displaystyle\varphi_{v}^{(w,z)}\left(\left[\begin{smallmatrix}t_{1}&0\\ 0&t_{2}\end{smallmatrix}\right]\,\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\,k\right)=|t_{1}/t_{2}|_{v}^{-w}\,\{A_{v}(w,z)\,h_{v}^{(w,z)}(x)+A_{v}(w,-z)\,h_{v}^{(w,-z)}(x)\},
(t1,t2Fv×,xFv,k𝕂v),formulae-sequencesubscript𝑡1subscript𝑡2superscriptsubscript𝐹𝑣formulae-sequence𝑥subscript𝐹𝑣𝑘subscript𝕂𝑣\displaystyle\qquad(t_{1},\,t_{2}\in F_{v}^{\times},\,x\in F_{v},\,k\in{\mathbb{K}}_{v}),

where

hv(w,z)(x)superscriptsubscript𝑣𝑤𝑧𝑥\displaystyle h_{v}^{(w,z)}(x) ={max(1,|x|v)(z+2w+1)/2,(vΣfin),(1+x2)(z+2w+1)/4F12(z+2w+14,z2w+14;z+22;1x2+1),(vΣ),\displaystyle=\begin{cases}\max(1,|x|_{v})^{-(z+2w+1)/2},\quad&(v\in\Sigma_{\rm fin}),\\ (1+x^{2})^{-(z+2w+1)/4}{}_{2}F_{1}\left(\tfrac{z+2w+1}{4},\tfrac{z-2w+1}{4};\tfrac{z+2}{2};\tfrac{1}{x^{2}+1}\right),\quad&(v\in\Sigma_{\infty}),\end{cases}
Av(w,z)subscript𝐴𝑣𝑤𝑧\displaystyle A_{v}(w,z) =ζFv(1)ζFv(z)ζFv((z+2w+1)/2)1ζFv((z2w+1)/2)1.absentsubscript𝜁subscript𝐹𝑣1subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣superscript𝑧2𝑤121subscript𝜁subscript𝐹𝑣superscript𝑧2𝑤121\displaystyle={\zeta_{F_{v}}(1)\,\zeta_{F_{v}}(-z)}{\zeta_{F_{v}}((-z+2w+1)/2)^{-1}\,\zeta_{F_{v}}((-z-2w+1)/2)^{-1}}.

The simple poles of the factor ζFv(z)subscript𝜁subscript𝐹𝑣𝑧\zeta_{F_{v}}(-z) of Av(w,z)subscript𝐴𝑣𝑤𝑧A_{v}(w,z) may cause singularities of φv(w,z)superscriptsubscript𝜑𝑣𝑤𝑧\varphi_{v}^{(w,z)}, but they are removable due to the obvious functional equation φv(w,z)(g)=φv(w,z)(g)superscriptsubscript𝜑𝑣𝑤𝑧𝑔superscriptsubscript𝜑𝑣𝑤𝑧𝑔\varphi_{v}^{(w,z)}(g)=\varphi_{v}^{(w,-z)}(g). For any vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}, we collect several easily proved formulas for later use:

(6.2) Av(0,z)+Av(0,z)=1,ζFv(z)ζFv(z+12)1+ζFv(z)ζFv(z+12)1=1,formulae-sequencesubscript𝐴𝑣0𝑧subscript𝐴𝑣0𝑧1subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣superscript𝑧121subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣superscript𝑧1211\displaystyle A_{v}(0,z)+A_{v}(0,-z)=1,\quad{\zeta_{F_{v}}(-z)}{\zeta_{F_{v}}\left(\tfrac{-z+1}{2}\right)^{-1}}+{\zeta_{F_{v}}(z)}{\zeta_{F_{v}}\left(\tfrac{z+1}{2}\right)^{-1}}=1,
(6.3) Av(0,z)qvz2ζFv(1z2)ζFv(1+z2)1+Av(0,z)qvz2ζFv(1+z2)ζFv(1z2)1=0.subscript𝐴𝑣0𝑧superscriptsubscript𝑞𝑣𝑧2subscript𝜁subscript𝐹𝑣1𝑧2subscript𝜁subscript𝐹𝑣superscript1𝑧21subscript𝐴𝑣0𝑧superscriptsubscript𝑞𝑣𝑧2subscript𝜁subscript𝐹𝑣1𝑧2subscript𝜁subscript𝐹𝑣superscript1𝑧210\displaystyle A_{v}(0,z)\,q_{v}^{\frac{z}{2}}{\zeta_{F_{v}}\left(\tfrac{1-z}{2}\right)}{\zeta_{F_{v}}\left(\tfrac{1+z}{2}\right)^{-1}}+A_{v}(0,-z)q_{v}^{\frac{-z}{2}}{\zeta_{F_{v}}\left(\tfrac{1+z}{2}\right)}{\zeta_{F_{v}}\left(\tfrac{1-z}{2}\right)^{-1}}=0.
Lemma 6.1.

The function φv(w,z)superscriptsubscript𝜑𝑣𝑤𝑧\varphi_{v}^{(w,z)} is the unique complex-valued smooth function on Gvsubscript𝐺𝑣G_{v} such that φv(w,z)(12)=1superscriptsubscript𝜑𝑣𝑤𝑧subscript121\varphi_{v}^{(w,z)}(1_{2})=1 which satisfies the following conditions:

(6.4) φv(w,z)([t100t2]gk)superscriptsubscript𝜑𝑣𝑤𝑧delimited-[]subscript𝑡100subscript𝑡2𝑔𝑘\displaystyle\varphi_{v}^{(w,z)}\left(\left[\begin{smallmatrix}t_{1}&0\\ 0&t_{2}\end{smallmatrix}\right]gk\right) =|t1/t2|vwφv(w,z)(g),t1,t2Fv×,gGv,k𝕂v,formulae-sequenceabsentsuperscriptsubscriptsubscript𝑡1subscript𝑡2𝑣𝑤superscriptsubscript𝜑𝑣𝑤𝑧𝑔subscript𝑡1formulae-sequencesubscript𝑡2superscriptsubscript𝐹𝑣formulae-sequence𝑔subscript𝐺𝑣𝑘subscript𝕂𝑣\displaystyle=|t_{1}/t_{2}|_{v}^{-w}\varphi_{v}^{(w,z)}(g),\quad t_{1},\,t_{2}\in F_{v}^{\times},\,g\in G_{v},\,k\in{\mathbb{K}}_{v},
(6.5) R(𝕋v)φv(w,z)𝑅subscript𝕋𝑣subscriptsuperscript𝜑𝑤𝑧𝑣\displaystyle R({\mathbb{T}}_{v})\varphi^{(w,z)}_{v} =qv1/2(qvz/2+qvz/2)φv(w,z)if vΣfin,absentsuperscriptsubscript𝑞𝑣12superscriptsubscript𝑞𝑣𝑧2superscriptsubscript𝑞𝑣𝑧2superscriptsubscript𝜑𝑣𝑤𝑧if vΣfin\displaystyle=q_{v}^{1/2}(q_{v}^{z/2}+q_{v}^{-z/2})\,\varphi_{v}^{(w,z)}\quad\text{if $v\in\Sigma_{\rm fin}$},
(6.6) R(Ωv)φv(w,z)𝑅subscriptΩ𝑣subscriptsuperscript𝜑𝑤𝑧𝑣\displaystyle R(\Omega_{v})\varphi^{(w,z)}_{v} =21(z21)φv(w,z)if vΣ,absentsuperscript21superscript𝑧21superscriptsubscript𝜑𝑣𝑤𝑧if vΣ\displaystyle=2^{-1}(z^{2}-1)\,\varphi_{v}^{(w,z)}\quad\text{if $v\in\Sigma_{\infty}$},

where ΩvsubscriptΩ𝑣\Omega_{v} for vΣ𝑣subscriptΣv\in\Sigma_{\infty} is the Casimir operator of Gvsubscript𝐺𝑣G_{v}.

Proof.

Let vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. In the same way as [30, Lemma 5.2], for a(m)=φv(w,z)([1ϖvm01])𝑎𝑚superscriptsubscript𝜑𝑣𝑤𝑧delimited-[]1subscriptsuperscriptitalic-ϖ𝑚𝑣01a(m)=\varphi_{v}^{(w,z)}\left(\left[\begin{smallmatrix}1&\varpi^{-m}_{v}\\ 0&1\end{smallmatrix}\right]\right) (m0(m\in{\mathbb{N}}_{0}), we deduce a recurrence relation

qv1+wa(m+1)+qvwa(m1)=qv1/2(qvz/2+qvz/2)a(m),(m>0),superscriptsubscript𝑞𝑣1𝑤𝑎𝑚1superscriptsubscript𝑞𝑣𝑤𝑎𝑚1superscriptsubscript𝑞𝑣12superscriptsubscript𝑞𝑣𝑧2superscriptsubscript𝑞𝑣𝑧2𝑎𝑚𝑚0\displaystyle q_{v}^{1+w}a(m+1)+q_{v}^{-w}a(m-1)=q_{v}^{1/2}(q_{v}^{z/2}+q_{v}^{-z/2})a(m),\quad(m>0),
(qv1)qvwa(1)+(qvw+qvw)a(0)=qv1/2(qvz/2+qvz/2)a(0),subscript𝑞𝑣1superscriptsubscript𝑞𝑣𝑤𝑎1superscriptsubscript𝑞𝑣𝑤superscriptsubscript𝑞𝑣𝑤𝑎0superscriptsubscript𝑞𝑣12superscriptsubscript𝑞𝑣𝑧2superscriptsubscript𝑞𝑣𝑧2𝑎0\displaystyle(q_{v}-1)q_{v}^{w}a(1)+(q_{v}^{w}+q_{v}^{-w})a(0)=q_{v}^{1/2}(q_{v}^{z/2}+q_{v}^{-z/2})a(0),

and a(0)=1𝑎01a(0)=1 from (6.5) and φ(w,z)(12)=1superscript𝜑𝑤𝑧subscript121\varphi^{(w,z)}(1_{2})=1, which can be solved uniquely by a(m)=Av(w,z)hv(w,z)(ϖvm)+Av(w,z)hv(w,z)(ϖvm)𝑎𝑚subscript𝐴𝑣𝑤𝑧superscriptsubscript𝑣𝑤𝑧superscriptsubscriptitalic-ϖ𝑣𝑚subscript𝐴𝑣𝑤𝑧superscriptsubscript𝑣𝑤𝑧superscriptsubscriptitalic-ϖ𝑣𝑚a(m)=A_{v}(w,z)\,h_{v}^{(w,z)}(\varpi_{v}^{-m})+A_{v}(w,-z)\,h_{v}^{(w,-z)}(\varpi_{v}^{-m}). By the Iwasawa decomposition and (6.4), we are done. Let vΣ𝑣subscriptΣv\in\Sigma_{\infty}. From [10, Proposition 4.3] for f(r)=φv(w,z)([coshrsinhrsinhrcoshr])𝑓𝑟superscriptsubscript𝜑𝑣𝑤𝑧delimited-[]𝑟𝑟𝑟𝑟f(r)=\varphi_{v}^{(w,z)}\left(\left[\begin{smallmatrix}{\cosh\,}r&{\sinh\,}r\\ {\sinh\,}r&{\cosh\,}r\end{smallmatrix}\right]\right), we have the differential equation

(12d2dr2+sinh 2rcosh 2rddr+2w2cosh22r)f(r)=z212f(r)12superscript𝑑2𝑑superscript𝑟22𝑟2𝑟𝑑𝑑𝑟2superscript𝑤2superscript22𝑟𝑓𝑟superscript𝑧212𝑓𝑟\left(\tfrac{1}{2}\tfrac{{{d}}^{2}}{{{d}}r^{2}}+\tfrac{{\sinh\,}2r}{{\cosh\,}2r}\tfrac{{{d}}}{{{d}}r}+\tfrac{2w^{2}}{{\cosh\,}^{2}2r}\right)f(r)=\tfrac{z^{2}-1}{2}f(r)

and f(0)=1𝑓01f(0)=1 from (6.6) and φ(w,z)(12)=1superscript𝜑𝑤𝑧subscript121\varphi^{(w,z)}(1_{2})=1, which has the unique Csuperscript𝐶C^{\infty} solution

f(r)=(cosh22r)(z+1)/4F12(z+2w+14,z2w+14;12;sinh22rcosh22r),r.formulae-sequence𝑓𝑟superscriptsuperscript22𝑟𝑧14subscriptsubscript𝐹12𝑧2𝑤14𝑧2𝑤1412superscript22𝑟superscript22𝑟𝑟f(r)=({\cosh\,}^{2}2r)^{-(z+1)/4}{}_{2}F_{1}\left(\tfrac{z+2w+1}{4},\tfrac{z-2w+1}{4};\tfrac{1}{2};\tfrac{{\sinh\,}^{2}2r}{{\cosh\,}^{2}2r}\right),\quad r\in{\mathbb{R}}.

By the Gauss connection formula (on the last line of [16, p.47]), it turns out that f(r)=(1+x2)w/2{Av(w,z)hv(w,z)(x)+Av(w,z)hv(w,z)(x)}𝑓𝑟superscript1superscript𝑥2𝑤2subscript𝐴𝑣𝑤𝑧superscriptsubscript𝑣𝑤𝑧𝑥subscript𝐴𝑣𝑤𝑧superscriptsubscript𝑣𝑤𝑧𝑥f(r)=(1+x^{2})^{w/2}\{A_{v}(w,z)\,h_{v}^{(w,z)}(x)+A_{v}(w,-z)\,h_{v}^{(w,-z)}(x)\} with 1+x2=cosh22r1superscript𝑥2superscript22𝑟1+x^{2}={\cosh\,}^{2}2r. By [30, Lemma 3.1], we are done. ∎

6.2. Unfolding and contour shifting

From Lemma 2.5 and Proposition 3.2,

Z𝔸GF\G𝔸𝑑gξHF\GFaF×{1}|Φ(𝐬;g1ξ1[a001]ξg)β(g)|subscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸differential-d𝑔subscript𝜉\subscript𝐻𝐹subscript𝐺𝐹subscript𝑎superscript𝐹1Φ𝐬superscript𝑔1superscript𝜉1delimited-[]𝑎001𝜉𝑔subscriptsuperscript𝛽𝑔\displaystyle\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}{{d}}g\,\sum_{\xi\in H_{F}\backslash G_{F}}\sum_{a\in F^{\times}-\{1\}}\left|\Phi\left({\mathbf{s}};g^{-1}\xi^{-1}\left[\begin{smallmatrix}a&0\\ 0&1\end{smallmatrix}\right]\xi g\right)\,{\mathcal{E}}^{*}_{\beta}(g)\right|
\displaystyle\leqslant Z𝔸GF\G𝔸𝑑gγZF\GF|Φ(𝐬;g1γg)β(g)|<+,subscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸differential-d𝑔subscript𝛾\subscript𝑍𝐹subscript𝐺𝐹Φ𝐬superscript𝑔1𝛾𝑔subscriptsuperscript𝛽𝑔\displaystyle\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}{{d}}g\,\sum_{\gamma\in Z_{F}\backslash G_{F}}\left|\Phi\left({\mathbf{s}};g^{-1}\gamma g\right)\,{\mathcal{E}}^{*}_{\beta}(g)\right|<+\infty,

which justifies the applications of Fubini’s theorem in the following computation:

𝕁hyp(𝐬,β)subscript𝕁hyp𝐬𝛽\displaystyle{\mathbb{J}}_{{\rm{hyp}}}({\mathbf{s}},\beta) =12Z𝔸HF\G𝔸𝑑gaF×{1}Φ(𝐬;g1[a001]g)β(g)absent12subscript\subscript𝑍𝔸subscript𝐻𝐹subscript𝐺𝔸differential-d𝑔subscript𝑎superscript𝐹1Φ𝐬superscript𝑔1delimited-[]𝑎001𝑔subscriptsuperscript𝛽𝑔\displaystyle=\tfrac{1}{2}\int_{Z_{\mathbb{A}}H_{F}\backslash G_{\mathbb{A}}}{{d}}g\,\sum_{a\in F^{\times}-\{1\}}\Phi\left({\mathbf{s}};g^{-1}\left[\begin{smallmatrix}a&0\\ 0&1\end{smallmatrix}\right]g\right)\,{\mathcal{E}}^{*}_{\beta}(g)
(6.7) =12aF×{1}𝔸𝑑x𝕂𝑑kΦ(𝐬;k1[a(a1)x01]k)Pβ(0;[1x01]),absent12subscript𝑎superscript𝐹1subscript𝔸differential-d𝑥subscript𝕂differential-d𝑘Φ𝐬superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘subscript𝑃𝛽0delimited-[]1𝑥01\displaystyle=\tfrac{1}{2}\sum_{a\in F^{\times}-\{1\}}\int_{{\mathbb{A}}}{{d}}x\int_{{\mathbb{K}}}{{d}}k\,\Phi\left({\mathbf{s}};k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)\,P_{\beta}\left(0;\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right),

where for w𝑤w\in{\mathbb{C}} and gG𝔸𝑔subscript𝐺𝔸g\in G_{\mathbb{A}}, we set

(6.8) Pβ(w;g)=F×\𝔸×β([t001]g)|t|𝔸wd×t,subscript𝑃𝛽𝑤𝑔subscript\superscript𝐹superscript𝔸superscriptsubscript𝛽delimited-[]𝑡001𝑔superscriptsubscript𝑡𝔸𝑤superscript𝑑𝑡\displaystyle P_{\beta}(w;g)=\int_{F^{\times}\backslash{\mathbb{A}}^{\times}}{\mathcal{E}}_{\beta}^{*}\left(\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]g\right)\,|t|_{\mathbb{A}}^{w}\,{{d}}^{\times}t,

which is shown to be absolutely convergent by Proposition 3.2 for all w𝑤w\in{\mathbb{C}}. By (3.1), the value β([t001][1x01])superscriptsubscript𝛽delimited-[]𝑡001delimited-[]1𝑥01{\mathcal{E}}_{\beta}^{*}\left(\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right) (t𝔸×,x𝔸)formulae-sequence𝑡superscript𝔸𝑥𝔸(t\in{\mathbb{A}}^{\times},\,x\in{\mathbb{A}}) is expressed as the sum of the following three contour integrals:

(6.9) Cβ±(t)superscriptsubscript𝐶𝛽plus-or-minus𝑡\displaystyle C_{\beta}^{\pm}(t) =Lσβ(z)ΛF(z)|t|𝔸(±z+1)/2𝑑z,absentsubscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹minus-or-plus𝑧superscriptsubscript𝑡𝔸plus-or-minus𝑧12differential-d𝑧\displaystyle=\textstyle{\int}_{L_{\sigma}}\beta(z)\Lambda_{F}(\mp z)|t|_{\mathbb{A}}^{(\pm z+1)/2}\,{{d}}z,
(6.10) 𝒲β(t,x)subscript𝒲𝛽𝑡𝑥\displaystyle{\mathcal{W}}_{\beta}(t,x) =Lσβ(z)ΛF(z)aF×Wψ(z;[at001][1x01])dz,absentsubscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧subscript𝑎superscript𝐹subscript𝑊𝜓𝑧delimited-[]𝑎𝑡001delimited-[]1𝑥01𝑑𝑧\displaystyle=\textstyle{\int}_{L_{\sigma}}\beta(z)\Lambda_{F}(-z)\,\sum_{a\in F^{\times}}W_{\psi}\left(z;\left[\begin{smallmatrix}at&0\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}z,

where 1<σ1𝜎1<\sigma. For any N1>0subscript𝑁10N_{1}>0, by shifting the contour Lσsubscript𝐿𝜎L_{\sigma} to Lσ1subscript𝐿subscript𝜎1L_{\sigma_{1}} (σ1=2N11subscript𝜎12subscript𝑁11\sigma_{1}=-2N_{1}-1), we have the inequality

|Cβ+(t)|Lσ1|β(z)ΛF(z)||dz|×|t|𝔸N1,superscriptsubscript𝐶𝛽𝑡subscriptsubscript𝐿subscript𝜎1𝛽𝑧subscriptΛ𝐹𝑧𝑑𝑧superscriptsubscript𝑡𝔸subscript𝑁1\displaystyle|C_{\beta}^{+}(t)|\leqslant\textstyle{\int}_{L_{\sigma_{1}}}|\beta(z)\Lambda_{F}(-z)|\,|{{d}}z|\times|t|_{\mathbb{A}}^{-N_{1}},

which yields the bound Cβ+(t)N1|t|𝔸N1subscriptmuch-less-thansubscript𝑁1superscriptsubscript𝐶𝛽𝑡superscriptsubscript𝑡𝔸subscript𝑁1C_{\beta}^{+}(t)\ll_{N_{1}}|t|_{\mathbb{A}}^{-N_{1}} for t𝔸×𝑡superscript𝔸t\in{\mathbb{A}}^{\times}. Combining this with the bound Cβ+(t)ϵ|t|𝔸N1subscriptmuch-less-thanitalic-ϵsuperscriptsubscript𝐶𝛽𝑡superscriptsubscript𝑡𝔸subscript𝑁1C_{\beta}^{+}(t)\ll_{\epsilon}|t|_{\mathbb{A}}^{N_{1}}, which is immediate from (6.9) with σ=2N11𝜎2subscript𝑁11\sigma=2N_{1}-1, we have

Cβ+(t)N1,ϵmin(|t|𝔸N1,|t|𝔸N1),t𝔸×.formulae-sequencesubscriptmuch-less-thansubscript𝑁1italic-ϵsuperscriptsubscript𝐶𝛽𝑡superscriptsubscript𝑡𝔸subscript𝑁1superscriptsubscript𝑡𝔸subscript𝑁1𝑡superscript𝔸C_{\beta}^{+}(t)\ll_{N_{1},\epsilon}\min(|t|_{\mathbb{A}}^{-N_{1}},|t|_{\mathbb{A}}^{N_{1}}),\quad t\in{\mathbb{A}}^{\times}.

From this, the integral

β+(w):=F×\𝔸×Cβ+(t)|t|𝔸wd×t=vol(F×\𝔸1)0Cβ+(t¯)twd×tassignsubscriptsuperscript𝛽𝑤subscript\superscript𝐹superscript𝔸superscriptsubscript𝐶𝛽𝑡superscriptsubscript𝑡𝔸𝑤superscript𝑑𝑡vol\superscript𝐹superscript𝔸1superscriptsubscript0superscriptsubscript𝐶𝛽¯𝑡superscript𝑡𝑤superscript𝑑𝑡{\mathfrak{C}}^{+}_{\beta}(w):=\textstyle{\int}_{F^{\times}\backslash{\mathbb{A}}^{\times}}C_{\beta}^{+}(t)\,|t|_{\mathbb{A}}^{w}\,{{d}}^{\times}t={\operatorname{vol}}(F^{\times}\backslash{\mathbb{A}}^{1})\,\textstyle{\int}_{0}^{\infty}C_{\beta}^{+}(\underline{t})\,t^{w}\,{{d}}^{\times}t

is seen to be absolutely convergent for all w𝑤w\in{\mathbb{C}}. In the same way as [30, Lemma 7.6], using the residue theorem for σ,σ1𝜎subscript𝜎1\sigma,\sigma_{1}\in\mathbb{R} such that σ1<2Re(w)1<σsubscript𝜎12Re𝑤1𝜎\sigma_{1}<-2\operatorname{Re}(w)-1<\sigma, we have

0Cβ+(t¯)twd×tsuperscriptsubscript0superscriptsubscript𝐶𝛽¯𝑡superscript𝑡𝑤superscript𝑑𝑡\displaystyle\textstyle{\int}_{0}^{\infty}C_{\beta}^{+}(\underline{t})\,t^{w}\,{{d}}^{\times}t =01d×tLσβ(z)ΛF(z)tw+(z+1)/2𝑑z+1d×tLσ1β(z)ΛF(z)tw+(z+1)/2𝑑zabsentsuperscriptsubscript01superscript𝑑𝑡subscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧superscript𝑡𝑤𝑧12differential-d𝑧superscriptsubscript1superscript𝑑𝑡subscriptsubscript𝐿subscript𝜎1𝛽𝑧subscriptΛ𝐹𝑧superscript𝑡𝑤𝑧12differential-d𝑧\displaystyle=\textstyle{\int}_{0}^{1}{{d}}^{\times}t\,\textstyle{\int}_{L_{\sigma}}\beta(z)\Lambda_{F}(-z)\,t^{w+(z+1)/2}\,{{d}}z+\textstyle{\int}_{1}^{\infty}{{d}}^{\times}t\,\textstyle{\int}_{L_{\sigma_{1}}}\beta(z)\Lambda_{F}(-z)t^{w+(z+1)/2}\,{{d}}z
=Lσβ(z)ΛF(z)dzw+(z+1)/2+Lσ1β(z)ΛF(z)dzw+(z+1)/2absentsubscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧𝑑𝑧𝑤𝑧12subscriptsubscript𝐿subscript𝜎1𝛽𝑧subscriptΛ𝐹𝑧𝑑𝑧𝑤𝑧12\displaystyle=\textstyle{\int}_{L_{\sigma}}\beta(z)\tfrac{\Lambda_{F}(-z){{d}}z}{w+(z+1)/2}+\textstyle{\int}_{L_{\sigma_{1}}}\beta(z)\tfrac{-\Lambda_{F}(-z){{d}}z}{w+(z+1)/2}
=2πiResz=2w1(β(z)ΛF(z)1w+(z+1)/2)absent2𝜋𝑖subscriptRes𝑧2𝑤1𝛽𝑧subscriptΛ𝐹𝑧1𝑤𝑧12\displaystyle=2\pi i{\rm{Res}}_{z=-2w-1}\left(\beta(z)\Lambda_{F}(-z)\tfrac{1}{w+(z+1)/2}\right)
=4πiβ(2w1)ΛF(1+2w).absent4𝜋𝑖𝛽2𝑤1subscriptΛ𝐹12𝑤\displaystyle=4\pi i\beta(-2w-1)\Lambda_{F}(1+2w).

In a similar manner as above, we have the estimation

Cβ(t)N1,ϵmin(|t|𝔸N1,|t|𝔸N1),t𝔸×formulae-sequencesubscriptmuch-less-thansubscript𝑁1italic-ϵsuperscriptsubscript𝐶𝛽𝑡superscriptsubscript𝑡𝔸subscript𝑁1superscriptsubscript𝑡𝔸subscript𝑁1𝑡superscript𝔸C_{\beta}^{-}(t)\ll_{N_{1},\epsilon}\min(|t|_{\mathbb{A}}^{-N_{1}},|t|_{\mathbb{A}}^{N_{1}}),\quad t\in{\mathbb{A}}^{\times}

for any N1>0subscript𝑁10N_{1}>0. When we shift the contour to the left to obtain the majorant |t|𝔸N1superscriptsubscript𝑡𝔸subscript𝑁1|t|_{\mathbb{A}}^{N_{1}}, we note that the singularity at z=0,1𝑧01z=0,1 of ΛF(z)subscriptΛ𝐹𝑧\Lambda_{F}(z) is canceled with the zeros of β(z)𝛽𝑧\beta(z). Hence the integral

β(w):=F×\𝔸×Cβ(t)|t|𝔸wd×tassignsubscriptsuperscript𝛽𝑤subscript\superscript𝐹superscript𝔸superscriptsubscript𝐶𝛽𝑡superscriptsubscript𝑡𝔸𝑤superscript𝑑𝑡{\mathfrak{C}}^{-}_{\beta}(w):=\textstyle{\int}_{F^{\times}\backslash{\mathbb{A}}^{\times}}C_{\beta}^{-}(t)\,|t|_{\mathbb{A}}^{w}\,{{d}}^{\times}t

is absolutely convergent for all w𝑤w\in{\mathbb{C}}, and is evaluated as 4πivol(F×\𝔸1)β(2w+1)ΛF(1+2w).4𝜋𝑖vol\superscript𝐹superscript𝔸1𝛽2𝑤1subscriptΛ𝐹12𝑤-4\pi i\,{\operatorname{vol}}(F^{\times}\backslash\mathbb{A}^{1})\beta(2w+1)\Lambda_{F}(1+2w). Consequently, β(w)+β+(w)=4πivol(F×\𝔸1)(β(2w1)β(2w+1))ΛF(1+2w)superscriptsubscript𝛽𝑤superscriptsubscript𝛽𝑤4𝜋𝑖vol\superscript𝐹superscript𝔸1𝛽2𝑤1𝛽2𝑤1subscriptΛ𝐹12𝑤{\mathfrak{C}}_{\beta}^{-}(w)+{\mathfrak{C}}_{\beta}^{+}(w)=4\pi i\,{\operatorname{vol}}(F^{\times}\backslash\mathbb{A}^{1})(\beta(-2w-1)-\beta(2w+1))\Lambda_{F}(1+2w). The absolute convergence of the integral

𝔚β(w;x)=F×\𝔸×𝒲β(t,x)|t|𝔸wd×tsubscript𝔚𝛽𝑤𝑥subscript\superscript𝐹superscript𝔸subscript𝒲𝛽𝑡𝑥superscriptsubscript𝑡𝔸𝑤superscript𝑑𝑡\displaystyle{\mathfrak{W}}_{\beta}(w;x)=\textstyle{\int}_{F^{\times}\backslash{\mathbb{A}}^{\times}}{\mathcal{W}}_{\beta}(t,x)\,|t|_{\mathbb{A}}^{w}\,{{d}}^{\times}t

for Re(w)>(Re(z)+1)/2Re𝑤Re𝑧12\operatorname{Re}(w)>(\operatorname{Re}(z)+1)/2 is confirmed by the inequality

F×\𝔸×|𝒲β(t,x)||t|𝔸Re(w)d×tLσ|β(z)ΛF(z)|(𝔸×|Wψ(z;[t001][1x01])||t|𝔸Re(w)d×t)|dz|subscript\superscript𝐹superscript𝔸subscript𝒲𝛽𝑡𝑥superscriptsubscript𝑡𝔸Re𝑤superscript𝑑𝑡subscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧subscriptsuperscript𝔸subscript𝑊𝜓𝑧delimited-[]𝑡001delimited-[]1𝑥01superscriptsubscript𝑡𝔸Re𝑤superscript𝑑𝑡𝑑𝑧\displaystyle\textstyle{\int}_{F^{\times}\backslash{\mathbb{A}}^{\times}}|{\mathcal{W}}_{\beta}(t,x)|\,|t|_{\mathbb{A}}^{\operatorname{Re}(w)}\,{{d}}^{\times}t\leqslant\textstyle{\int}_{L_{\sigma}}\left|\beta(z)\Lambda_{F}(-z)\right|\ \left(\int_{{\mathbb{A}}^{\times}}|W_{\psi}\left(z;\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)|\,|t|_{\mathbb{A}}^{\operatorname{Re}(w)}{{d}}^{\times}t\right)\,|{{d}}z|

combined with the bound

𝔸×|Wψ(z;[t001][1x01])||t|𝔸Re(w)d×t=O(1),zLσ,Re(w)>(σ+1)/2,formulae-sequencesubscriptsuperscript𝔸subscript𝑊𝜓𝑧delimited-[]𝑡001delimited-[]1𝑥01superscriptsubscript𝑡𝔸Re𝑤superscript𝑑𝑡𝑂1formulae-sequence𝑧subscript𝐿𝜎Re𝑤𝜎12\displaystyle\textstyle{\int}_{{\mathbb{A}}^{\times}}|W_{\psi}\left(z;\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)||t|_{\mathbb{A}}^{\operatorname{Re}(w)}\,{{d}}^{\times}t=O(1),\quad z\in L_{\sigma},\,\operatorname{Re}(w)>(\sigma+1)/2,

which follows from (3.2) and (3.3). By Lemma 6.1, on the region Re(w)>(|Re(z)|+1)/2Re𝑤Re𝑧12\operatorname{Re}(w)>(|\operatorname{Re}(z)|+1)/2, for all g=(gv)G𝔸𝑔subscript𝑔𝑣subscript𝐺𝔸g=(g_{v})\in G_{\mathbb{A}}, we have

𝔸×Wψ(z;[t001]g)|t|𝔸wd×t=DFz/2+w1/2ζF(w+z+12)ζF(w+z+12)ζF(z+1)vΣFφv(w,z)(gv).subscriptsuperscript𝔸subscript𝑊𝜓𝑧delimited-[]𝑡001𝑔superscriptsubscript𝑡𝔸𝑤superscript𝑑𝑡superscriptsubscript𝐷𝐹𝑧2𝑤12subscript𝜁𝐹𝑤𝑧12subscript𝜁𝐹𝑤𝑧12subscript𝜁𝐹𝑧1subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣𝑤𝑧subscript𝑔𝑣\displaystyle\int_{{\mathbb{A}}^{\times}}W_{\psi}\left(z;\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]g\right)\,|t|_{\mathbb{A}}^{w}\,{{d}}^{\times}t=D_{F}^{-z/2+w-1/2}\frac{\zeta_{F}\left(w+\frac{z+1}{2}\right)\zeta_{F}\left(w+\frac{-z+1}{2}\right)}{\zeta_{F}(z+1)}\,\prod_{v\in\Sigma_{F}}\varphi_{v}^{(w,z)}\left(g_{v}\right).
Lemma 6.2.

For all w𝑤w\in\mathbb{C} such that 0Re(w)10Re𝑤10\leqslant\operatorname{Re}(w)\leqslant 1 and x=(xv)𝔸𝑥subscript𝑥𝑣𝔸x=(x_{v})\in{\mathbb{A}}, we have

Pβ(w;[1x01])=subscript𝑃𝛽𝑤delimited-[]1𝑥01absent\displaystyle P_{\beta}\left(w;\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)= Lσβ(z)DFwζF(w+z+12)ζF(w+z+12)vΣFφv(w,z)([1xv01])dzsubscriptsubscript𝐿superscript𝜎𝛽𝑧superscriptsubscript𝐷𝐹𝑤subscript𝜁𝐹𝑤𝑧12subscript𝜁𝐹𝑤𝑧12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣𝑤𝑧delimited-[]1subscript𝑥𝑣01𝑑𝑧\displaystyle\int_{L_{\sigma^{\prime}}}\beta(z)\,D_{F}^{w}\,\zeta_{F}\left(w+\tfrac{z+1}{2}\right)\zeta_{F}\left(w+\tfrac{-z+1}{2}\right)\,\prod_{v\in\Sigma_{F}}\varphi_{v}^{(w,z)}\left(\left[\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}z
+4πiRess=1ζF(s)β(2w1)ζF(2w)vΣFφv(w,2w1)(([1xv01]))4𝜋𝑖subscriptRes𝑠1subscript𝜁𝐹𝑠𝛽2𝑤1subscript𝜁𝐹2𝑤subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣𝑤2𝑤1delimited-[]1subscript𝑥𝑣01\displaystyle+4\pi i\,\operatorname{Res}_{s=1}\zeta_{F}(s)\,\beta(2w-1)\zeta_{F}(2w)\prod_{v\in\Sigma_{F}}\varphi_{v}^{(w,2w-1)}(\left(\left[\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right]\right))
+4πivol(F×\𝔸1)ΛF(1+2w)(β(2w1)β(2w+1)),(σ>1).4𝜋𝑖vol\superscript𝐹superscript𝔸1subscriptΛ𝐹12𝑤𝛽2𝑤1𝛽2𝑤1superscript𝜎1\displaystyle+4\pi i\,{\operatorname{vol}}(F^{\times}\backslash\mathbb{A}^{1})\Lambda_{F}(1+2w)(\beta(-2w-1)-\beta(2w+1)),\quad(\sigma^{\prime}>1).
Proof.

Let F(z,w)𝐹𝑧𝑤F(z,w) denote the integrand of the integral on the right-hand side. Suppose σ>1𝜎1\sigma>1 and Re(w)>σ+12Re𝑤𝜎12\operatorname{Re}(w)>\frac{\sigma+1}{2}. By taking σ>1superscript𝜎1\sigma^{\prime}>1 so that σ>σsuperscript𝜎𝜎\sigma^{\prime}>\sigma and σ12<Re(w)<σ+12superscript𝜎12Re𝑤superscript𝜎12\frac{\sigma^{\prime}-1}{2}<\operatorname{Re}(w)<\frac{\sigma^{\prime}+1}{2}, Cauchy’s integral theorem yields

LσF(z,w)𝑑z=LσF(z,w)𝑑z2πiResz=2w1F(z,w).subscriptsubscript𝐿𝜎𝐹𝑧𝑤differential-d𝑧subscriptsubscript𝐿superscript𝜎𝐹𝑧𝑤differential-d𝑧2𝜋𝑖subscriptRes𝑧2𝑤1𝐹𝑧𝑤\displaystyle\int_{L_{\sigma}}F(z,w)dz=\int_{L_{\sigma^{\prime}}}F(z,w){{d}}z-2\pi i\operatorname{Res}_{z=2w-1}F(z,w).

Hence, we have the expression of Pβ(w;[1x01])subscript𝑃𝛽𝑤delimited-[]1𝑥01P_{\beta}\left(w;\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right) in the assertion for σ12<Re(w)<σ+12superscript𝜎12Re𝑤superscript𝜎12\frac{\sigma^{\prime}-1}{2}<\operatorname{Re}(w)<\frac{\sigma^{\prime}+1}{2}. We remark that the first term LσF(z,w)𝑑zsubscriptsubscript𝐿superscript𝜎𝐹𝑧𝑤differential-d𝑧\int_{L_{\sigma^{\prime}}}F(z,w){{d}}z is holomorphic on σ12<Re(w)<σ+12superscript𝜎12Re𝑤superscript𝜎12\frac{\sigma^{\prime}-1}{2}<\operatorname{Re}(w)<\frac{\sigma^{\prime}+1}{2} by Re(w+z+12)>1Re𝑤𝑧121\operatorname{Re}(w+\frac{z+1}{2})>1 and 0<Re(z+z+12)<10Re𝑧𝑧1210<\operatorname{Re}(z+\frac{-z+1}{2})<1, and that the second and the third terms are entire due to β(0)=β(±1)=β(±1)=0𝛽0𝛽plus-or-minus1superscript𝛽plus-or-minus10\beta(0)=\beta(\pm 1)=\beta^{\prime}(\pm 1)=0. ∎

By shifting the contour Lσsubscript𝐿superscript𝜎L_{\sigma^{\prime}} to Lσsubscript𝐿𝜎L_{\sigma} (1<σ<11𝜎1-1<\sigma<1) after substituting z=0𝑧0z=0, from Lemma 6.2, we have the expression

(6.11) Pβ(0;[1x01])subscript𝑃𝛽0delimited-[]1𝑥01\displaystyle P_{\beta}\left(0;\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right) =Lσβ(z)ζF(z+12)ζF(z+12)vΣFφv(0,z)([1xv01])dz,(1<σ<1).absentsubscriptsubscript𝐿𝜎𝛽𝑧subscript𝜁𝐹𝑧12subscript𝜁𝐹𝑧12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣0𝑧delimited-[]1subscript𝑥𝑣01𝑑𝑧1𝜎1\displaystyle=\int_{L_{\sigma}}\beta(z)\,\zeta_{F}\left(\tfrac{z+1}{2}\right)\zeta_{F}\left(\tfrac{-z+1}{2}\right)\,\prod_{v\in\Sigma_{F}}\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}z,\quad(-1<\sigma<1).

6.3. The orbital integrals

By substituting the expression (6.11) for the last formula of (6.7) and formally changing the order of the integral and the summation, the integral

(6.12) 𝔉(z)(a)=𝔸(𝕂Φ(𝐬,k1[a(a1)x01]k)𝑑k)vΣFφv(0,z)([1xv01])dx,aF{0,1}formulae-sequencesuperscript𝔉𝑧𝑎subscript𝔸subscript𝕂Φ𝐬superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘differential-d𝑘subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣0𝑧delimited-[]1subscript𝑥𝑣01𝑑𝑥𝑎𝐹01\displaystyle{\mathfrak{F}}^{(z)}(a)=\int_{{\mathbb{A}}}\left(\int_{{\mathbb{K}}}\Phi\left({\mathbf{s}},k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)\,{{d}}k\right)\,\prod_{v\in\Sigma_{F}}\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x,\quad a\in F-\{0,1\}

emerges naturally. In this subsection, we prove the following proposition, which is sufficient to legitimatize the formal computation explained above.

Proposition 6.3.

Suppose minvSRe(sv)>1subscript𝑣𝑆Resubscript𝑠𝑣1\min_{v\in S}\operatorname{Re}(s_{v})>1.

  • (1)

    Let aF{0,1}𝑎𝐹01a\in F-\{0,1\}. The integral (6.12) converges absolutely for Re(z)<2l¯1Re𝑧2¯𝑙1\operatorname{Re}(z)<2{\underline{l}}-1. On that region, we have the product formula 𝔉(z)(a)=vΣF𝔉v(z)(a)superscript𝔉𝑧𝑎subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}^{(z)}(a)=\prod_{v\in\Sigma_{F}}{\mathfrak{F}}_{v}^{(z)}(a) with

    𝔉v(z)(a)=Fv(𝕂vΦv(kv1[a(a1)xv01]kv)𝑑kv)φv(0,z)([1xv01])𝑑xv,vΣF,formulae-sequencesuperscriptsubscript𝔉𝑣𝑧𝑎subscriptsubscript𝐹𝑣subscriptsubscript𝕂𝑣subscriptΦ𝑣superscriptsubscript𝑘𝑣1delimited-[]𝑎𝑎1subscript𝑥𝑣01subscript𝑘𝑣differential-dsubscript𝑘𝑣superscriptsubscript𝜑𝑣0𝑧delimited-[]1subscript𝑥𝑣01differential-dsubscript𝑥𝑣𝑣subscriptΣ𝐹{\mathfrak{F}}_{v}^{(z)}(a)=\int_{F_{v}}\left(\int_{{\mathbb{K}}_{v}}\Phi_{v}\left(k_{v}^{-1}\left[\begin{smallmatrix}a&(a-1)x_{v}\\ 0&1\end{smallmatrix}\right]k_{v}\right)\,{{d}}k_{v}\right)\,\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x_{v},\quad v\in\Sigma_{F},

    where Φv(gv)subscriptΦ𝑣subscript𝑔𝑣\Phi_{v}(g_{v}) is the v𝑣v-th factor of Φ(𝐬;g)Φ𝐬𝑔\Phi({\mathbf{s}};g). We have 𝔉(z)(a)=0superscript𝔉𝑧𝑎0{\mathfrak{F}}^{(z)}(a)=0 unless aF+𝑎superscriptsubscript𝐹a\in F_{\infty}^{+}, where F+={aF|av>0(vΣ)}superscriptsubscript𝐹conditional-setsubscript𝑎subscript𝐹subscript𝑎𝑣0𝑣subscriptΣF_{\infty}^{+}=\{a_{\infty}\in F_{\infty}|\,a_{v}>0\,(v\in\Sigma_{\infty})\,\}.

  • (2)

    Set

    σ0=min({2lv1|vΣ}{2Re(sv)1|vS}).subscript𝜎0conditional-set2subscript𝑙𝑣1𝑣subscriptΣconditional-set2Resubscript𝑠𝑣1𝑣𝑆\sigma_{0}=\min(\{2l_{v}-1|\,v\in\Sigma_{\infty}\}\cup\{2\operatorname{Re}(s_{v})-1|\,v\in S\,\}).

    For any σ(0,σ0)𝜎0subscript𝜎0\sigma\in(0,\sigma_{0}) and ϵ(0,1)italic-ϵ01\epsilon\in(0,1), there exists a positive number Cϵsubscript𝐶italic-ϵC_{\epsilon} independent of z𝑧z and 𝐬𝐬{\mathbf{s}} such that

    |𝔉(z)(a)|Cϵ{vΣfinϕv(a)}{vΣ(1+|a|v)lv2+σ+12}superscript𝔉𝑧𝑎subscript𝐶italic-ϵsubscriptproduct𝑣subscriptΣfinsubscriptitalic-ϕ𝑣𝑎subscriptproduct𝑣subscriptΣsuperscript1subscript𝑎𝑣subscript𝑙𝑣2𝜎12\displaystyle|{\mathfrak{F}}^{(z)}(a)|\leqslant C_{\epsilon}\,\{\prod_{v\in\Sigma_{\rm fin}}\phi_{v}(a)\}\,\{\prod_{v\in\Sigma_{\infty}}(1+|a|_{v})^{-\frac{l_{v}}{2}+\frac{\sigma+1}{2}}\}

    for all aF+(F×{1})𝑎superscriptsubscript𝐹superscript𝐹1a\in F_{\infty}^{+}\cap(F^{\times}-\{1\}) and |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma, where

    ϕv(a)={δ(a𝔬v×),(vΣfin(SS(𝔫))),4(1+qv)1δ(a𝔬v×)(1+qv1/2)δ(a1+𝔭v){ordv(a1)+1},(vS(𝔫)),{max(0,ordv(a1))+1}max(1,|a1|v)|Re(z)|Re(sv)2+ϵ,(vS).\displaystyle\phi_{v}(a)=\begin{cases}\delta(a\in\mathfrak{o}_{v}^{\times}),\quad&(v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))),\\ {4}{(1+q_{v})^{-1}}\delta(a\in\mathfrak{o}_{v}^{\times})(1+q_{v}^{1/2})^{\delta(a\in 1+{\mathfrak{p}}_{v})}\,\{{\rm{ord}}_{v}(a-1)+1\},\quad&(v\in S({\mathfrak{n}})),\\ \{\max(0,{\rm{ord}}_{v}(a-1))+1\}\,\max(1,|a-1|_{v})^{\frac{|\operatorname{Re}(z)|-\operatorname{Re}(s_{v})}{2}+\epsilon},\quad&(v\in S).\end{cases}

For the proof, we use the following explicit formulas of the local integrals 𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a), which are proved later in §10.

Theorem 6.4.

Let vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F} and aFv×{1}𝑎superscriptsubscript𝐹𝑣1a\in F_{v}^{\times}-\{1\}.

  1. (1)

    Suppose vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Then, on the region |Re(z)|<2lv1Re𝑧2subscript𝑙𝑣1|\operatorname{Re}(z)|<2l_{v}-1, we have

    𝔉v(z)(a)=superscriptsubscript𝔉𝑣𝑧𝑎absent\displaystyle{\mathfrak{F}}_{v}^{(z)}(a)= 4πΓ(lv)Γ(lv+z12)Γ(lv+z12)Γ(1+z2)Γ(1z2)δ(a>0)|a|v1/2|a1|v1𝔓z121lv(λv(a)1/2)4𝜋Γsubscript𝑙𝑣Γsubscript𝑙𝑣𝑧12Γsubscript𝑙𝑣𝑧12subscriptΓ1𝑧2subscriptΓ1𝑧2𝛿𝑎0superscriptsubscript𝑎𝑣12superscriptsubscript𝑎1𝑣1superscriptsubscript𝔓𝑧121subscript𝑙𝑣subscript𝜆𝑣superscript𝑎12\displaystyle\frac{4\pi}{\Gamma(l_{v})}\,\frac{\Gamma\left(l_{v}+\tfrac{z-1}{2}\right)\Gamma\left(l_{v}+\tfrac{-z-1}{2}\right)}{\Gamma_{\mathbb{R}}\left(\tfrac{1+z}{2}\right)\Gamma_{\mathbb{R}}\left(\tfrac{1-z}{2}\right)}\,\delta(a>0)\,|a|_{v}^{1/2}|a-1|_{v}^{-1}{\mathfrak{P}}_{\frac{z-1}{2}}^{1-l_{v}}\left(\lambda_{v}(a)^{-1/2}\right)

    with λv(a)=|(a1)/(a+1)|v2subscript𝜆𝑣𝑎superscriptsubscript𝑎1𝑎1𝑣2\lambda_{v}(a)=|(a-1)/(a+1)|_{v}^{2}. Here 𝔓νμ(x)superscriptsubscript𝔓𝜈𝜇𝑥{\mathfrak{P}}_{\nu}^{\mu}(x) is the Legendre function of the 1st kind which is defined for points x𝑥x\in{\mathbb{C}} outside the interval (,+1]1(-\infty,+1] of the real axis ([16, §4.1]).

  2. (2)

    Let vΣfin(SS(𝔫))𝑣subscriptΣfin𝑆𝑆𝔫v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}})). Then, we have

    𝔉v(z)(a)=δ(a𝔬v×)qvdv/2𝒪v,01,(z)((a1)2a).superscriptsubscript𝔉𝑣𝑧𝑎𝛿𝑎superscriptsubscript𝔬𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscript𝒪𝑣01𝑧superscript𝑎12𝑎\displaystyle{\mathfrak{F}}_{v}^{(z)}(a)=\delta(a\in\mathfrak{o}_{v}^{\times})\,q_{v}^{-d_{v}/2}\,{\mathcal{O}}_{v,0}^{1,(z)}((a-1)^{-2}a).

    In particular, 𝔉v(z)(a)=qvdv/2superscriptsubscript𝔉𝑣𝑧𝑎superscriptsubscript𝑞𝑣subscript𝑑𝑣2{\mathfrak{F}}_{v}^{(z)}(a)=q_{v}^{-d_{v}/2} if |a|v=|a1|v=1subscript𝑎𝑣subscript𝑎1𝑣1|a|_{v}=|a-1|_{v}=1.

  3. (3)

    Suppose vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}). Then, we have

    𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎\displaystyle{\mathfrak{F}}_{v}^{(z)}(a) =δ(a𝔬v×)qvdv/2𝒪v,11,(z)((a1)2a).absent𝛿𝑎superscriptsubscript𝔬𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscript𝒪𝑣11𝑧superscript𝑎12𝑎\displaystyle=\delta(a\in\mathfrak{o}_{v}^{\times})\,q_{v}^{-d_{v}/2}\,{\mathcal{O}}_{v,1}^{1,(z)}((a-1)^{-2}a).
  4. (4)

    Suppose vS𝑣𝑆v\in S. Then, for 2Re(sv)+1>|Re(z)|2Resubscript𝑠𝑣1Re𝑧2\operatorname{Re}(s_{v})+1>|\operatorname{Re}(z)| we have

    𝔉v(z)(sv;a)=superscriptsubscript𝔉𝑣𝑧subscript𝑠𝑣𝑎absent\displaystyle{\mathfrak{F}}_{v}^{(z)}(s_{v};a)= qvdv2𝒮v1,(z)((a1)2a).superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscript𝒮𝑣1𝑧superscript𝑎12𝑎\displaystyle q_{v}^{-\frac{d_{v}}{2}}{\mathcal{S}}_{v}^{1,(z)}((a-1)^{-2}a).
Lemma 6.5.

We have the inequality

|𝔉v(z)(a)|3|ordv(a1)||a1|v|Re(z)|+12superscriptsubscript𝔉𝑣𝑧𝑎3subscriptord𝑣𝑎1superscriptsubscript𝑎1𝑣Re𝑧12|{\mathfrak{F}}_{v}^{(z)}(a)|\leqslant 3\,|{\rm{ord}}_{v}(a-1)|\,|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}}

for all a1+𝔭v𝑎1subscript𝔭𝑣a\in 1+{\mathfrak{p}}_{v} and z𝑧z\in{\mathbb{C}}.

Proof.

We suppose a1+𝔭v𝑎1subscript𝔭𝑣a\in 1+{\mathfrak{p}}_{v} and set t=qvz/2𝑡superscriptsubscript𝑞𝑣𝑧2t=q_{v}^{z/2} and α=ordv(a1)𝛼subscriptord𝑣𝑎1\alpha=\operatorname{ord}_{v}(a-1)\in{\mathbb{N}}. Then, by Theorem 6.4 (2), 𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a) is the product of qvdv/2|a1|v1/2superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscript𝑎1𝑣12q_{v}^{-d_{v}/2}|a-1|_{v}^{-1/2} and

f(t)𝑓𝑡\displaystyle f(t) :=1q1/2t1t2tα+1qv1/2t11t2tα=tα{1+g(t)t2α(1tqv1/2)}assignabsent1superscript𝑞12𝑡1superscript𝑡2superscript𝑡𝛼1superscriptsubscript𝑞𝑣12superscript𝑡11superscript𝑡2superscript𝑡𝛼superscript𝑡𝛼1𝑔𝑡superscript𝑡2𝛼1𝑡superscriptsubscript𝑞𝑣12\displaystyle:=\tfrac{1-q^{-1/2}t}{1-t^{2}}t^{-\alpha}+\tfrac{1-q_{v}^{-1/2}t^{-1}}{1-t^{-2}}t^{\alpha}=t^{\alpha}\left\{1+{g(t)}\,t^{-2\alpha}(1-tq_{v}^{-1/2})\right\}

with g(t)=j=0α1t2j𝑔𝑡superscriptsubscript𝑗0𝛼1superscript𝑡2𝑗g(t)=\sum_{j=0}^{\alpha-1}t^{2j}. From the second expression, f(t)𝑓𝑡f(t) is seen to be holomorphic away from t=0𝑡0t=0. Let |t|1𝑡1|t|\geqslant 1; then |g(t)||t|2αα|t|2𝑔𝑡superscript𝑡2𝛼𝛼superscript𝑡2|g(t)||t|^{-2\alpha}\leqslant\alpha|t|^{-2} and

|f(t)||t|α𝑓𝑡superscript𝑡𝛼\displaystyle{|f(t)|}{|t|^{-\alpha}} 1+g(t)|t|2α(1+|t|qv1/2)1+α(|t|2+qv1/2|t|1)3α.absent1𝑔𝑡superscript𝑡2𝛼1𝑡superscriptsubscript𝑞𝑣121𝛼superscript𝑡2superscriptsubscript𝑞𝑣12superscript𝑡13𝛼\displaystyle\leqslant 1+g(t)\,|t|^{-2\alpha}(1+|t|q_{v}^{-1/2})\leqslant 1+\alpha(|t|^{-2}+q_{v}^{-1/2}|t|^{-1})\leqslant 3\alpha.

Thus |f(t)|3α|t|α𝑓𝑡3𝛼superscript𝑡𝛼|f(t)|\leqslant 3\alpha\,|t|^{\alpha} for |t|1𝑡1|t|\geqslant 1. From this we have |f(t)|3α|t|α𝑓𝑡3𝛼superscript𝑡𝛼|f(t)|\leqslant 3\alpha\,|t|^{-\alpha} by the obvious functional equation f(t1)=f(t)𝑓superscript𝑡1𝑓𝑡f(t^{-1})=f(t). ∎

Corollary 6.6.

Let aF×{1}𝑎superscript𝐹1a\in F^{\times}-\{1\}. Given ϵ>0italic-ϵ0\epsilon>0, we have a constant Cϵ>0subscript𝐶italic-ϵ0C_{\epsilon}>0 such that

vΣfin(SS(𝔫))|𝔉v(z)(a)|δ(a𝔬(SS(𝔫))×)CϵvΣfin(SS(𝔫))|a1|v|Re(z)|+12ϵsubscriptproduct𝑣subscriptΣfin𝑆𝑆𝔫superscriptsubscript𝔉𝑣𝑧𝑎𝛿𝑎𝔬superscript𝑆𝑆𝔫subscript𝐶italic-ϵsubscriptproduct𝑣subscriptΣfin𝑆𝑆𝔫superscriptsubscript𝑎1𝑣Re𝑧12italic-ϵ\displaystyle\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}|{\mathfrak{F}}_{v}^{(z)}(a)|\leqslant\delta(a\in\mathfrak{o}(S\cup S({\mathfrak{n}}))^{\times})\,C_{\epsilon}\,\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}-\epsilon}

for aF×{1}𝑎superscript𝐹1a\in F^{\times}-\{1\} and z𝑧z\in{\mathbb{C}}, where 𝔬(SS(𝔫))𝔬𝑆𝑆𝔫\mathfrak{o}(S\cup S({\mathfrak{n}})) is the ring of SS(𝔫)𝑆𝑆𝔫S\cup S({\mathfrak{n}})-integers in F𝐹F.

Proof.

Choose Aϵ>1subscript𝐴italic-ϵ1A_{\epsilon}>1 such that 3xAϵ 2ϵx3𝑥subscript𝐴italic-ϵsuperscript2italic-ϵ𝑥3x\leqslant A_{\epsilon}\,2^{\epsilon x} for all x1𝑥1x\geqslant 1. We fix q(ϵ)>1𝑞italic-ϵ1q(\epsilon)>1 such that Aϵ 2ϵqϵsubscript𝐴italic-ϵsuperscript2italic-ϵsuperscript𝑞italic-ϵA_{\epsilon}\,2^{\epsilon}\leqslant q^{\epsilon} for all q>q(ϵ)𝑞𝑞italic-ϵq>q(\epsilon), and set P(ϵ)={vΣfin|qvq(ϵ)}𝑃italic-ϵconditional-set𝑣subscriptΣfinsubscript𝑞𝑣𝑞italic-ϵP(\epsilon)=\{v\in\Sigma_{\rm fin}|\,q_{v}\leqslant q(\epsilon)\}. Then 3|ordv(a1)|Aϵ|a1|vϵ3subscriptord𝑣𝑎1subscript𝐴italic-ϵsuperscriptsubscript𝑎1𝑣italic-ϵ3|{\rm{ord}}_{v}(a-1)|\leqslant A_{\epsilon}\,|a-1|_{v}^{-\epsilon} for all a1+𝔭v𝑎1subscript𝔭𝑣a\in 1+{\mathfrak{p}}_{v} with vP(ϵ)𝑣𝑃italic-ϵv\in P(\epsilon), and 3|ordv(a1)||a1|vϵ3subscriptord𝑣𝑎1superscriptsubscript𝑎1𝑣italic-ϵ3|{\rm{ord}}_{v}(a-1)|\leqslant|a-1|_{v}^{-\epsilon} for all a1+𝔭v𝑎1subscript𝔭𝑣a\in 1+{\mathfrak{p}}_{v} with vΣfinP(ϵ)𝑣subscriptΣfin𝑃italic-ϵv\in\Sigma_{\rm fin}-P(\epsilon). Setting Cϵ=Aϵ#P(ϵ)subscript𝐶italic-ϵsuperscriptsubscript𝐴italic-ϵ#𝑃italic-ϵC_{\epsilon}=A_{\epsilon}^{\#P(\epsilon)}, we are done by Lemma 6.5. ∎

Lemma 6.7.

We have the estimate

|𝔉v(z)(a)|4(1+qv)1δ(a𝔬v×)(1+qv1/2)δ(a1+𝔭v)(ordv(a1)+1)|a1|v|Re(z)|+12superscriptsubscript𝔉𝑣𝑧𝑎4superscript1subscript𝑞𝑣1𝛿𝑎superscriptsubscript𝔬𝑣superscript1superscriptsubscript𝑞𝑣12𝛿𝑎1subscript𝔭𝑣subscriptord𝑣𝑎11superscriptsubscript𝑎1𝑣Re𝑧12\displaystyle|{\mathfrak{F}}_{v}^{(z)}(a)|\leqslant 4(1+q_{v})^{-1}\,\delta(a\in\mathfrak{o}_{v}^{\times})\,(1+q_{v}^{1/2})^{\delta(a\in 1+{\mathfrak{p}}_{v})}\,({\rm{ord}}_{v}(a-1)+1)\,|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}}

for vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}), aFv×{1}𝑎superscriptsubscript𝐹𝑣1a\in F_{v}^{\times}-\{1\} and z𝑧z\in{\mathbb{C}}.

Proof.

Let vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}), a1+𝔭v𝑎1subscript𝔭𝑣a\in 1+{\mathfrak{p}}_{v}. Set t=qvz/2𝑡superscriptsubscript𝑞𝑣𝑧2t=q_{v}^{z/2}, α=ordv(a1)1𝛼subscriptord𝑣𝑎11\alpha={\rm{ord}}_{v}(a-1)\geqslant 1 and

f(t)𝑓𝑡\displaystyle f(t) =1qv1/2t1t2(1+qv1/2t)tα+1qv1/2t11t2(1+qv1/2t1)tαabsent1superscriptsubscript𝑞𝑣12𝑡1superscript𝑡21superscriptsubscript𝑞𝑣12𝑡superscript𝑡𝛼1superscriptsubscript𝑞𝑣12superscript𝑡11superscript𝑡21superscriptsubscript𝑞𝑣12superscript𝑡1superscript𝑡𝛼\displaystyle=\tfrac{1-q_{v}^{-1/2}t}{1-t^{2}}(1+q_{v}^{1/2}t)t^{-\alpha}+\tfrac{1-q_{v}^{-1/2}t^{-1}}{1-t^{-2}}(1+q_{v}^{1/2}t^{-1})t^{\alpha}
=tα{1+t2+t1(qv1/2qv1/2)+t2αg(t)(1qv1/2t)(1+qv1/2t)}absentsuperscript𝑡𝛼1superscript𝑡2superscript𝑡1superscriptsubscript𝑞𝑣12superscriptsubscript𝑞𝑣12superscript𝑡2𝛼𝑔𝑡1superscriptsubscript𝑞𝑣12𝑡1superscriptsubscript𝑞𝑣12𝑡\displaystyle=t^{\alpha}\{1+t^{-2}+t^{-1}(q_{v}^{1/2}-q_{v}^{-1/2})+t^{-2\alpha}g(t)(1-q_{v}^{-1/2}t)(1+q_{v}^{1/2}t)\}

with g(t)=j=0α2t2j𝑔𝑡superscriptsubscript𝑗0𝛼2superscript𝑡2𝑗g(t)=\sum_{j=0}^{\alpha-2}t^{2j}. Then Theorem 6.4 (3) shows that 𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a) is the product of qvdv/21+qv|a1|v1/2superscriptsubscript𝑞𝑣subscript𝑑𝑣21subscript𝑞𝑣superscriptsubscript𝑎1𝑣12\frac{q_{v}^{-d_{v}/2}}{1+q_{v}}|a-1|_{v}^{-1/2} and f(t)𝑓𝑡f(t). If |t|1𝑡1|t|\geqslant 1, we have |g(t)||t|2α|t|4j=0α2|t|2jα|t|4𝑔𝑡superscript𝑡2𝛼superscript𝑡4superscriptsubscript𝑗0𝛼2superscript𝑡2𝑗𝛼superscript𝑡4|g(t)||t|^{-2\alpha}\leqslant|t|^{-4}\sum_{j=0}^{\alpha-2}|t|^{-2j}\leqslant\alpha|t|^{-4} and

|f(t)|𝑓𝑡\displaystyle|f(t)| |t|α{1+|t|2+|t|1(qv1/2+qv1/2)+|t|2α|g(t)|(1+qv1/2|t|)(1+qv1/2|t|)}absentsuperscript𝑡𝛼1superscript𝑡2superscript𝑡1superscriptsubscript𝑞𝑣12superscriptsubscript𝑞𝑣12superscript𝑡2𝛼𝑔𝑡1superscriptsubscript𝑞𝑣12𝑡1superscriptsubscript𝑞𝑣12𝑡\displaystyle\leqslant|t|^{\alpha}\{1+|t|^{-2}+|t|^{-1}(q_{v}^{1/2}+q_{v}^{-1/2})+|t|^{-2\alpha}|g(t)|(1+q_{v}^{-1/2}|t|)(1+q_{v}^{1/2}|t|)\}
|t|α{1+1+(1+qv1/2)+2α(1+1)(1+qv1/2)}4(α+1)(1+qv1/2)|t|α.absentsuperscript𝑡𝛼111superscriptsubscript𝑞𝑣122𝛼111superscriptsubscript𝑞𝑣124𝛼11superscriptsubscript𝑞𝑣12superscript𝑡𝛼\displaystyle\leqslant|t|^{\alpha}\{1+1+(1+q_{v}^{1/2})+2\alpha(1+1)(1+q_{v}^{1/2})\}\leqslant 4(\alpha+1)(1+q_{v}^{1/2})\,|t|^{\alpha}.

Hence |f(t)|4(α+1)(1+qv1/2)|t|α𝑓𝑡4𝛼11superscriptsubscript𝑞𝑣12superscript𝑡𝛼|f(t)|\leqslant 4(\alpha+1)(1+q_{v}^{1/2})|t|^{\alpha} for |t|1𝑡1|t|\geqslant 1. By the obvious functional equation f(t)=f(t1)𝑓𝑡𝑓superscript𝑡1f(t)=f(t^{-1}), we also have |f(t)|4(α+1)(1+qv1/2)|t|α𝑓𝑡4𝛼11superscriptsubscript𝑞𝑣12superscript𝑡𝛼|f(t)|\leqslant 4(\alpha+1)(1+q_{v}^{1/2})|t|^{-\alpha} for |t|1𝑡1|t|\leqslant 1. Thus

|𝔉v(z)(a)|qvdv/2(1+qv)1 4(ordv(a1)+1)(1+qv1/2)|a1|v|Re(z)|+12.superscriptsubscript𝔉𝑣𝑧𝑎superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscript1subscript𝑞𝑣14subscriptord𝑣𝑎111superscriptsubscript𝑞𝑣12superscriptsubscript𝑎1𝑣Re𝑧12|{\mathfrak{F}}_{v}^{(z)}(a)|\leqslant q_{v}^{-d_{v}/2}(1+q_{v})^{-1}\,4(\,{\rm{ord}}_{v}(a-1)+1)(1+q_{v}^{1/2})|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}}.

Lemma 6.8.

Let vS𝑣𝑆v\in S. On the region Re(sv)>1Resubscript𝑠𝑣1\operatorname{Re}(s_{v})>1 and |Re(z)|<2Re(sv)1Re𝑧2Resubscript𝑠𝑣1|\operatorname{Re}(z)|<2\operatorname{Re}(s_{v})-1,

|𝔉v(z)(sv;a)|16qvRe(sv)+12(1+ordv(a1))|a1|v|Re(z)|+12for |a1|v<1,superscriptsubscript𝔉𝑣𝑧subscript𝑠𝑣𝑎16superscriptsubscript𝑞𝑣Resubscript𝑠𝑣121subscriptord𝑣𝑎1superscriptsubscript𝑎1𝑣Re𝑧12for |a1|v<1\displaystyle|{\mathfrak{F}}_{v}^{(z)}(s_{v};a)|\leqslant 16q_{v}^{-\frac{\operatorname{Re}(s_{v})+1}{2}}(1+{\rm{ord}}_{v}(a-1))|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}}\quad\text{for $|a-1|_{v}<1$},
|𝔉v(z)(sv;a)|16qvRe(sv)+12|a1|v(Re(sv)+1)|a|vRe(sv)+12for |a1|v1.superscriptsubscript𝔉𝑣𝑧subscript𝑠𝑣𝑎16superscriptsubscript𝑞𝑣Resubscript𝑠𝑣12superscriptsubscript𝑎1𝑣Resubscript𝑠𝑣1superscriptsubscript𝑎𝑣Resubscript𝑠𝑣12for |a1|v1\displaystyle|{\mathfrak{F}}_{v}^{(z)}(s_{v};a)|\leqslant 16q_{v}^{-\frac{\operatorname{Re}(s_{v})+1}{2}}|a-1|_{v}^{-(\operatorname{Re}(s_{v})+1)}|a|_{v}^{\frac{\operatorname{Re}(s_{v})+1}{2}}\quad\text{for $|a-1|_{v}\geqslant 1$}.
Proof.

Put s=sv𝑠subscript𝑠𝑣s=s_{v}. Suppose |a1|v<1subscript𝑎1𝑣1|a-1|_{v}<1 and let ν=ordv(a1)𝜈subscriptord𝑣𝑎1\nu={\rm{ord}}_{v}(a-1) and t=qvz/2𝑡superscriptsubscript𝑞𝑣𝑧2t=q_{v}^{z/2}. From Theorem 6.4 (4), a computation shows the identity 𝔉v(z)(a)=qvdv+s+12|a1|v1/2f(t)superscriptsubscript𝔉𝑣𝑧𝑎superscriptsubscript𝑞𝑣subscript𝑑𝑣𝑠12superscriptsubscript𝑎1𝑣12𝑓𝑡{\mathfrak{F}}_{v}^{(z)}(a)=q_{v}^{-\frac{d_{v}+s+1}{2}}|a-1|_{v}^{-1/2}\,f(t) with

f(t)𝑓𝑡\displaystyle f(t) =tν{t2νg(t)(1qv1/2t(1+qvs)+t2qvs1)+1+t2t1qv1/2(1+qvs)}(1qvs1/2t)(1qvs1/2t1),absentsuperscript𝑡𝜈superscript𝑡2𝜈𝑔𝑡1superscriptsubscript𝑞𝑣12𝑡1superscriptsubscript𝑞𝑣𝑠superscript𝑡2superscriptsubscript𝑞𝑣𝑠11superscript𝑡2superscript𝑡1superscriptsubscript𝑞𝑣121superscriptsubscript𝑞𝑣𝑠1superscriptsubscript𝑞𝑣𝑠12𝑡1superscriptsubscript𝑞𝑣𝑠12superscript𝑡1\displaystyle=\frac{t^{\nu}\{t^{-2\nu}g(t)(1-q_{v}^{-1/2}t(1+q_{v}^{-s})+t^{2}q_{v}^{-s-1})+1+t^{-2}-t^{-1}q_{v}^{-1/2}(1+q_{v}^{-s})\}}{(1-q_{v}^{-s-1/2}t)(1-q_{v}^{-s-1/2}t^{-1})},

where g(t)=j=0ν2t2j𝑔𝑡superscriptsubscript𝑗0𝜈2superscript𝑡2𝑗g(t)=\sum_{j=0}^{\nu-2}t^{2j}. Suppose |t|1𝑡1|t|\geqslant 1. Then |g(t)|<|t|2ν2ν𝑔𝑡superscript𝑡2𝜈2𝜈|g(t)|<|t|^{2\nu-2}\nu. From Re(s)>1Re𝑠1\operatorname{Re}(s)>1 and |Re(z)|<2Re(s)1Re𝑧2Re𝑠1|\operatorname{Re}(z)|<2\operatorname{Re}(s)-1, we have |qvs|<qv1<1superscriptsubscript𝑞𝑣𝑠superscriptsubscript𝑞𝑣11|q_{v}^{-s}|<q_{v}^{-1}<1 and |qvs1/2t±1|<qv1superscriptsubscript𝑞𝑣𝑠12superscript𝑡plus-or-minus1superscriptsubscript𝑞𝑣1|q_{v}^{-s-1/2}t^{\pm 1}|<q_{v}^{-1}. By these,

|f(t)|𝑓𝑡\displaystyle|f(t)| |t|ν{|t|2ν(1+qv1/2|t|(1+|qvs|)+|t|2|qvs|)+1+|t|2+|t|1(1+|qvs|)qv1/2}(1qv1)2absentsuperscript𝑡𝜈superscript𝑡2𝜈1superscriptsubscript𝑞𝑣12𝑡1superscriptsubscript𝑞𝑣𝑠superscript𝑡2superscriptsubscript𝑞𝑣𝑠1superscript𝑡2superscript𝑡11superscriptsubscript𝑞𝑣𝑠superscriptsubscript𝑞𝑣12superscript1superscriptsubscript𝑞𝑣12\displaystyle\leqslant{|t|^{\nu}\bigl{\{}|t|^{-2}\nu(1+q_{v}^{-1/2}|t|(1+|q_{v}^{-s}|)+|t|^{2}|q_{v}^{-s}|)+1+|t|^{-2}+|t|^{-1}(1+|q_{v}^{-s}|)q_{v}^{-1/2}\bigr{\}}}{(1-q_{v}^{-1})^{-2}}
|t|ν(4ν+4)(1qv1)24|t|ν(ν+1)(1/2)2=16|t|ν(ν+1).absentsuperscript𝑡𝜈4𝜈4superscript1superscriptsubscript𝑞𝑣124superscript𝑡𝜈𝜈1superscript12216superscript𝑡𝜈𝜈1\displaystyle\leqslant{|t|^{\nu}(4\nu+4)}{(1-q_{v}^{-1})^{-2}}\leqslant{4|t|^{\nu}(\nu+1)}{(1/2)^{-2}}=16|t|^{\nu}(\nu+1).

Thus |f(t)|16(ν+1)|t|ν𝑓𝑡16𝜈1superscript𝑡𝜈|f(t)|\leqslant 16(\nu+1)|t|^{\nu} for |t|1𝑡1|t|\geqslant 1. Since f(t)=f(t1)𝑓𝑡𝑓superscript𝑡1f(t)=f(t^{-1}), we then have |f(t)|16(ν+1)|t|ν𝑓𝑡16𝜈1superscript𝑡𝜈|f(t)|\leqslant 16(\nu+1)|t|^{-\nu} for |t|1𝑡1|t|\leqslant 1.

Suppose |a1|v1subscript𝑎1𝑣1|a-1|_{v}\geqslant 1 and set t=qvz/2𝑡superscriptsubscript𝑞𝑣𝑧2t=q_{v}^{z/2}. Then

|𝔉(z)(s;a)|superscript𝔉𝑧𝑠𝑎\displaystyle|{\mathfrak{F}}^{(z)}(s;a)| =qvdv+Re(s)+121+|qvs1||(1qvs1/2t)(1qvs1/2t1)||a1|vRe(s)1|a|vRe(s)+12absentsuperscriptsubscript𝑞𝑣subscript𝑑𝑣Re𝑠121superscriptsubscript𝑞𝑣𝑠11superscriptsubscript𝑞𝑣𝑠12𝑡1superscriptsubscript𝑞𝑣𝑠12superscript𝑡1superscriptsubscript𝑎1𝑣Re𝑠1superscriptsubscript𝑎𝑣Re𝑠12\displaystyle=q_{v}^{-\frac{d_{v}+\operatorname{Re}(s)+1}{2}}\frac{1+|q_{v}^{-s-1}|}{|(1-q_{v}^{-s-1/2}t)(1-q_{v}^{-s-1/2}t^{-1})|}\,|a-1|_{v}^{-\operatorname{Re}(s)-1}|a|_{v}^{\frac{\operatorname{Re}(s)+1}{2}}
qvRe(s)+121+qv2(1qv1)2|a1|vRe(s)1|a|vRe(s)+125qvRe(s)+12|a1|vRe(s)1|a|vRe(s)+12.absentsuperscriptsubscript𝑞𝑣Re𝑠121superscriptsubscript𝑞𝑣2superscript1superscriptsubscript𝑞𝑣12superscriptsubscript𝑎1𝑣Re𝑠1superscriptsubscript𝑎𝑣Re𝑠125superscriptsubscript𝑞𝑣Re𝑠12superscriptsubscript𝑎1𝑣Re𝑠1superscriptsubscript𝑎𝑣Re𝑠12\displaystyle\leqslant q_{v}^{-\frac{\operatorname{Re}(s)+1}{2}}\frac{1+q_{v}^{-2}}{(1-q_{v}^{-1})^{2}}|a-1|_{v}^{-\operatorname{Re}(s)-1}|a|_{v}^{\frac{\operatorname{Re}(s)+1}{2}}\leqslant 5q_{v}^{-\frac{\operatorname{Re}(s)+1}{2}}|a-1|_{v}^{-\operatorname{Re}(s)-1}|a|_{v}^{\frac{\operatorname{Re}(s)+1}{2}}.

Corollary 6.9.

On the region |Re(z)|<2Re(sv)1Re𝑧2Resubscript𝑠𝑣1|\operatorname{Re}(z)|<2\operatorname{Re}(s_{v})-1, Re(sv)>1Resubscript𝑠𝑣1\operatorname{Re}(s_{v})>1, we have

|𝔉v(z)(a)|16{max(0,ordv(a1))+1}max(1,|a1|v)|Re(z)|Re(sv)2|a1|v|Re(z)|+12\displaystyle|{\mathfrak{F}}^{(z)}_{v}(a)|\leqslant 16\{\max(0,{\rm{ord}}_{v}(a-1))+1\}\,\max(1,|a-1|_{v})^{\frac{|\operatorname{Re}(z)|-\operatorname{Re}(s_{v})}{2}}\,|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}}

for all aFv×{1}𝑎superscriptsubscript𝐹𝑣1a\in F_{v}^{\times}-\{1\}.

Proof.

If |a1|v1subscript𝑎1𝑣1|a-1|_{v}\geqslant 1, then |a|v|a1|v+12|a1|vsubscript𝑎𝑣subscript𝑎1𝑣12subscript𝑎1𝑣|a|_{v}\leqslant|a-1|_{v}+1\leqslant 2|a-1|_{v}. By this remark, we can deduce the inequality from those in Lemma 6.8. ∎

Lemma 6.10.

Let vΣ𝑣subscriptΣv\in\Sigma_{\infty}. On the region |Re(z)|<2lv1Re𝑧2subscript𝑙𝑣1|\operatorname{Re}(z)|<2l_{v}-1, the integral 𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a) converges absolutely and is a holomorphic function in z𝑧z for any fixed aF{0,1}𝑎𝐹01a\in F-\{0,1\}. We have

|𝔉v(z)(a)|δ(av>0)|a|vlv/2|a1|vlv,aF{0,1}formulae-sequencemuch-less-thansuperscriptsubscript𝔉𝑣𝑧𝑎𝛿subscript𝑎𝑣0superscriptsubscript𝑎𝑣subscript𝑙𝑣2superscriptsubscript𝑎1𝑣subscript𝑙𝑣𝑎𝐹01\displaystyle|{\mathfrak{F}}_{v}^{(z)}(a)|\ll\delta(a_{v}>0)\,|a|_{v}^{l_{v}/2}\,|a-1|_{v}^{-l_{v}},\quad a\in F-\{0,1\}

uniformly for z𝑧z lying in a compact set of |Re(z)|<2lv1Re𝑧2subscript𝑙𝑣1|\operatorname{Re}(z)|<2l_{v}-1.

Proof.

Recall Φv=ΦvlvsubscriptΦ𝑣superscriptsubscriptΦ𝑣subscript𝑙𝑣\Phi_{v}=\Phi_{v}^{l_{v}} is given by (2.1). By the Cartan decomposition, we have

Φvlv(k1[a(a1)x01]k)=δ(a>0) 2lvalv/2{(a+1)i(a1)x}lvsuperscriptsubscriptΦ𝑣subscript𝑙𝑣superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘𝛿𝑎0superscript2subscript𝑙𝑣superscript𝑎subscript𝑙𝑣2superscript𝑎1𝑖𝑎1𝑥subscript𝑙𝑣\Phi_{v}^{l_{v}}\left(k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)=\delta(a>0)\,2^{l_{v}}a^{l_{v}/2}\,\{(a+1)-i(a-1)x\}^{-l_{v}}

for any k𝕂v𝑘subscript𝕂𝑣k\in{\mathbb{K}}_{v}, a×𝑎superscripta\in{\mathbb{R}}^{\times} and x𝑥x\in{\mathbb{R}}. From now on, we suppose a>0𝑎0a>0 and a1𝑎1a\neq 1. We have 𝔉v(z)(a)=δ(a>0)2lvalv/2(I+(z,a)+I(z,a))superscriptsubscript𝔉𝑣𝑧𝑎𝛿𝑎0superscript2subscript𝑙𝑣superscript𝑎subscript𝑙𝑣2superscript𝐼𝑧𝑎superscript𝐼𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a)=\delta(a>0)2^{l_{v}}a^{l_{v}/2}(I^{+}(z,a)+I^{-}(z,a)), where

Iε(z,a)=Jε{(a+1)i(a1)x}lvφv(0,z)([1x01])𝑑xsuperscript𝐼𝜀𝑧𝑎subscriptsuperscript𝐽𝜀superscript𝑎1𝑖𝑎1𝑥subscript𝑙𝑣superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥\displaystyle I^{\varepsilon}(z,a)=\textstyle{\int}_{J^{\varepsilon}}\{(a+1)-i(a-1)x\}^{-l_{v}}\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x

with J+=[1,1]superscript𝐽11J^{+}=[-1,1], J=J+superscript𝐽superscript𝐽J^{-}={\mathbb{R}}-J^{+}. We examine the convergence of these integrals separately. By the obvious estimate |(a+1)i(a1)x|={(a+1)2+x2(a1)2}1/2|a+1|𝑎1𝑖𝑎1𝑥superscriptsuperscript𝑎12superscript𝑥2superscript𝑎1212𝑎1|(a+1)-i(a-1)x|=\{(a+1)^{2}+x^{2}(a-1)^{2}\}^{1/2}\geqslant|a+1|,

|I+(z,a)|superscript𝐼𝑧𝑎\displaystyle|I^{+}(z,a)| 11|(a+1)i(a1)x|lv|φv(0,z)([1x01])|𝑑x|a+1|lv11|φv(0,z)([1x01])|𝑑xabsentsuperscriptsubscript11superscript𝑎1𝑖𝑎1𝑥subscript𝑙𝑣superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥superscript𝑎1subscript𝑙𝑣superscriptsubscript11superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥\displaystyle\leqslant\textstyle{\int}_{-1}^{1}|(a+1)-i(a-1)x|^{-l_{v}}|\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)|\,{{d}}x\leqslant|a+1|^{-l_{v}}\textstyle{\int}_{-1}^{1}|\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)|\,{{d}}x

Since the integral is convergent for all z𝑧z\in{\mathbb{C}}, we have the estimate |I+(z,a)||a+1|lvmuch-less-thansuperscript𝐼𝑧𝑎superscript𝑎1subscript𝑙𝑣|I^{+}(z,a)|\ll|a+1|^{-l_{v}} compact uniformly in z𝑧z. In the same way, by the estimate |(a+1)i(a1)x||x||a1|𝑎1𝑖𝑎1𝑥𝑥𝑎1|(a+1)-i(a-1)x|\geqslant|x||a-1|,

|I(z,a)||a1|lv|x|>1|x|lv|φv(0,z)([1x01])|𝑑x.superscript𝐼𝑧𝑎superscript𝑎1subscript𝑙𝑣subscript𝑥1superscript𝑥subscript𝑙𝑣superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥\displaystyle|I^{-}(z,a)|\leqslant|a-1|^{-l_{v}}\textstyle{\int}_{|x|>1}|x|^{-l_{v}}|\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)|\,{{d}}x.

From the defining formula (6.1), we easily have the estimate φv(0,z)([1x01])=O((1+x2)(|Re(z)|1)/4)superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01𝑂superscript1superscript𝑥2Re𝑧14\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)=O((1+x^{2})^{(|\operatorname{Re}(z)|-1)/4}) as x𝑥x\rightarrow\infty. From this the integral in the majorant of |I(z,a)|superscript𝐼𝑧𝑎|I^{-}(z,a)| is seen to be convergent for |Re(z)|<2lv1Re𝑧2subscript𝑙𝑣1|\operatorname{Re}(z)|<2l_{v}-1. ∎

Corollary 6.11.

Let vΣ𝑣subscriptΣv\in\Sigma_{\infty} and 0<σ<2lv10𝜎2subscript𝑙𝑣10<\sigma<2l_{v}-1. We have the estimate

|𝔉v(z)(a)|δ(av>0)|a1|v|Re(z)|+12(1+|a|v)lv2+|Re(z)|+12,aF{0,1}formulae-sequencemuch-less-thansuperscriptsubscript𝔉𝑣𝑧𝑎𝛿subscript𝑎𝑣0superscriptsubscript𝑎1𝑣Re𝑧12superscript1subscript𝑎𝑣subscript𝑙𝑣2Re𝑧12𝑎𝐹01|{\mathfrak{F}}_{v}^{(z)}(a)|\ll\delta(a_{v}>0)\,|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}}\,(1+|a|_{v})^{-\frac{l_{v}}{2}+\frac{|\operatorname{Re}(z)|+1}{2}},\quad a\in F-\{0,1\}

uniformly for z𝑧z such that |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma.

Proof.

Let U𝑈U be a neighborhood of 111 in +subscript{\mathbb{R}}_{+} such that |a1|v<1subscript𝑎1𝑣1|a-1|_{v}<1 for all aU𝑎𝑈a\in U. From Lemma 6.10, we have the estimate |𝔉(z)(a)|δ(av>0)(1+|a|v)lv/2much-less-thansuperscript𝔉𝑧𝑎𝛿subscript𝑎𝑣0superscript1subscript𝑎𝑣subscript𝑙𝑣2|{\mathfrak{F}}^{(z)}(a)|\ll\delta(a_{v}>0)\,(1+|a|_{v})^{-l_{v}/2} on aU𝑎𝑈a\in{\mathbb{R}}-U uniformly in |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma. To have a bound on U𝑈U, we use Theorem 6.4 (1). By Stirling’s formula, Γ(lv+z12)Γ(lv+z12)Γ(1+z2)Γ(1z2)Γsubscript𝑙𝑣𝑧12Γsubscript𝑙𝑣𝑧12subscriptΓ1𝑧2subscriptΓ1𝑧2\frac{\Gamma\left(l_{v}+\tfrac{z-1}{2}\right)\Gamma\left(l_{v}+\tfrac{-z-1}{2}\right)}{\Gamma_{\mathbb{R}}\left(\tfrac{1+z}{2}\right)\Gamma_{\mathbb{R}}\left(\tfrac{1-z}{2}\right)} is vertically bounded on |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma. By the formula [16, line 12, p.184],

𝔓z121lv(λv(a)1/2)superscriptsubscript𝔓𝑧121subscript𝑙𝑣subscript𝜆𝑣superscript𝑎12\displaystyle{\mathfrak{P}}_{\frac{z-1}{2}}^{1-l_{v}}(\lambda_{v}(a)^{-1/2}) =1πΓ(lv1/2)|a|vlv12|a1|vz+120π{2avcosθ+av+1}z+12lv(sin2θ)lv1𝑑θ.absent1𝜋Γsubscript𝑙𝑣12superscriptsubscript𝑎𝑣subscript𝑙𝑣12superscriptsubscript𝑎1𝑣𝑧12superscriptsubscript0𝜋superscript2subscript𝑎𝑣𝜃subscript𝑎𝑣1𝑧12subscript𝑙𝑣superscriptsuperscript2𝜃subscript𝑙𝑣1differential-d𝜃\displaystyle=\tfrac{1}{\sqrt{\pi}\Gamma(l_{v}-1/2)}|a|_{v}^{\frac{l_{v}-1}{2}}|a-1|_{v}^{\frac{-z+1}{2}}\,\textstyle{\int}_{0}^{\pi}\{2\sqrt{a_{v}}\cos\theta+a_{v}+1\}^{\frac{z+1}{2}-l_{v}}(\sin^{2}\theta)^{l_{v}-1}\,{{d}}\theta.

For varying aU𝑎𝑈a\in U and |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma, the last integral is bounded by a constant. Thus, |𝔓z121lv(λv(a)1/2)|=O(|a1|v|Re(z)|+12)superscriptsubscript𝔓𝑧121subscript𝑙𝑣subscript𝜆𝑣superscript𝑎12𝑂superscriptsubscript𝑎1𝑣Re𝑧12|{\mathfrak{P}}_{\frac{z-1}{2}}^{1-l_{v}}(\lambda_{v}(a)^{-1/2})|=O(|a-1|_{v}^{\frac{-|\operatorname{Re}(z)|+1}{2}}) for aU𝑎𝑈a\in U and |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma. ∎

Now, Proposition 6.3 follows from Corollaries 6.6, 6.11 and 6.9 and Lemma 6.7. We use the product formula vΣF|a1|v=1subscriptproduct𝑣subscriptΣ𝐹subscript𝑎1𝑣1\prod_{v\in\Sigma_{F}}|a-1|_{v}=1 for aF×{1}𝑎superscript𝐹1a\in F^{\times}-\{1\} to eliminate the factors |a1|v|Re(z)|+12superscriptsubscript𝑎1𝑣Re𝑧12|a-1|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}} (vΣF)𝑣subscriptΣ𝐹(v\in\Sigma_{F}) in the majorant.

6.4. Proof of absolute convergence

Proposition 6.12.

Suppose Re(sv)>2Resubscript𝑠𝑣2\operatorname{Re}(s_{v})>2 for all vS𝑣𝑆v\in S. Let 0<σ<lv30𝜎subscript𝑙𝑣30<\sigma<l_{v}-3. The series aF×{1}𝔉(z)(a)subscript𝑎superscript𝐹1superscript𝔉𝑧𝑎\sum_{a\in F^{\times}-\{1\}}{\mathfrak{F}}^{(z)}(a) converges absolutely and uniformly for |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma and locally uniformly in 𝐬𝐬{\mathbf{s}} defining a holomorphic function of z𝑧z on the region

|Re(z)|<min({lv3|vΣ}{Re(sv)2|vS}).Re𝑧conditional-setsubscript𝑙𝑣3𝑣subscriptΣconditional-setResubscript𝑠𝑣2𝑣𝑆|\operatorname{Re}(z)|<\min\,(\{l_{v}-3|v\in\Sigma_{\infty}\}\cup\{\operatorname{Re}(s_{v})-2|\,v\in S\,\}).
Proof.

Let S0Σfinsubscript𝑆0subscriptΣfinS_{0}\subset\Sigma_{\rm fin} be such that Φv=chZv𝕂vsubscriptΦ𝑣subscriptchsubscript𝑍𝑣subscript𝕂𝑣\Phi_{v}={\rm ch}_{Z_{v}{\mathbb{K}}_{v}} and dv=0subscript𝑑𝑣0d_{v}=0 for all vΣfinS0𝑣subscriptΣfinsubscript𝑆0v\in\Sigma_{\rm fin}-S_{0}. Then we may take ϕv=ch𝔬v×subscriptitalic-ϕ𝑣subscriptchsuperscriptsubscript𝔬𝑣\phi_{v}={\rm ch}_{\mathfrak{o}_{v}^{\times}} in Proposition 6.3, by which the proof is reduced to showing the convergence of aF×{1}f(a)subscript𝑎superscript𝐹1𝑓𝑎\sum_{a\in F^{\times}-\{1\}}f(a) with f𝑓f being a function on the adeles defined as

f(a)={vΣfinϕv(av)}{vΣ(1+|av|v)lv/2+(σ+1)/2},a𝔸{0,1}.formulae-sequence𝑓𝑎subscriptproduct𝑣subscriptΣfinsubscriptitalic-ϕ𝑣subscript𝑎𝑣subscriptproduct𝑣subscriptΣsuperscript1subscriptsubscript𝑎𝑣𝑣subscript𝑙𝑣2𝜎12𝑎𝔸01\displaystyle f(a)=\{\prod_{v\in\Sigma_{\rm fin}}\phi_{v}(a_{v})\}\,\,\{\prod_{v\in\Sigma_{\infty}}(1+|a_{v}|_{v})^{-l_{v}/2+(\sigma+1)/2}\},\quad a\in{\mathbb{A}}-\{0,1\}.

Let 𝔬(S0)𝔬subscript𝑆0\mathfrak{o}(S_{0}) be the ring of S0subscript𝑆0S_{0}-integers in F𝐹F, i.e., 𝔬(S0)=FvΣfinS0𝔬v×vS0ΣFv𝔬subscript𝑆0𝐹subscriptproduct𝑣subscriptΣfinsubscript𝑆0subscript𝔬𝑣subscriptproduct𝑣subscript𝑆0subscriptΣsubscript𝐹𝑣\mathfrak{o}(S_{0})=F\cap\prod_{v\in\Sigma_{\rm fin}-S_{0}}\mathfrak{o}_{v}\times\prod_{v\in S_{0}\cup\Sigma_{\infty}}F_{v}. Set fv(xv)=ϕv(xv)subscript𝑓𝑣subscript𝑥𝑣subscriptitalic-ϕ𝑣subscript𝑥𝑣f_{v}(x_{v})=\phi_{v}(x_{v}) for vS0𝑣subscript𝑆0v\in S_{0} and fv(xv)=(1+|xv|v)lv/2+(σ+1)/2subscript𝑓𝑣subscript𝑥𝑣superscript1subscriptsubscript𝑥𝑣𝑣subscript𝑙𝑣2𝜎12f_{v}(x_{v})=(1+|x_{v}|_{v})^{-l_{v}/2+(\sigma+1)/2} if vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Let 𝒩vϖv𝔬vsubscript𝒩𝑣subscriptitalic-ϖ𝑣subscript𝔬𝑣{\mathcal{N}}_{v}\subset\varpi_{v}\mathfrak{o}_{v} be a compact neighborhood of 00 in Fvsubscript𝐹𝑣F_{v} such that ϕvsubscriptitalic-ϕ𝑣\phi_{v} is constant on cosets x+𝒩v𝑥subscript𝒩𝑣x+{\mathcal{N}}_{v} if vS0𝑣subscript𝑆0v\in S_{0}. Then fv(x+y)=fv(x)(xFv,y𝒩v)subscript𝑓𝑣𝑥𝑦subscript𝑓𝑣𝑥formulae-sequence𝑥subscript𝐹𝑣𝑦subscript𝒩𝑣f_{v}(x+y)=f_{v}(x)\,(x\in F_{v},\,y\in{\mathcal{N}}_{v}) if vS0𝑣subscript𝑆0v\in S_{0} and fv(x)fv(x+y)(xFv,y𝒩v)much-less-thansubscript𝑓𝑣𝑥subscript𝑓𝑣𝑥𝑦formulae-sequence𝑥subscript𝐹𝑣𝑦subscript𝒩𝑣f_{v}(x)\ll f_{v}(x+y)\,(x\in F_{v},\,y\in{\mathcal{N}}_{v}) if vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Hence if we set 𝒩S0,=vS0Σ𝒩vsubscript𝒩subscript𝑆0subscriptproduct𝑣subscript𝑆0subscriptΣsubscript𝒩𝑣{\mathcal{N}}_{S_{0},\infty}=\prod_{v\in S_{0}\cup\Sigma_{\infty}}{\mathcal{N}}_{v}, then f(x)f(x+y)(xFS0,,y𝒩S0,)much-less-than𝑓𝑥𝑓𝑥𝑦formulae-sequence𝑥subscript𝐹subscript𝑆0𝑦subscript𝒩subscript𝑆0f(x)\ll f(x+y)\,(x\in F_{S_{0},\infty},\,y\in{\mathcal{N}}_{S_{0},\infty}). Since 𝔬(S0)𝔬subscript𝑆0\mathfrak{o}(S_{0}) is a discrete subgroup of FS0,=vS0ΣFvsubscript𝐹subscript𝑆0subscriptproduct𝑣subscript𝑆0subscriptΣsubscript𝐹𝑣F_{S_{0},\infty}=\prod_{v\in S_{0}\cup\Sigma_{\infty}}F_{v}, by choosing 𝒩S0,subscript𝒩subscript𝑆0{\mathcal{N}}_{S_{0},\infty} small enough, we have

aF×{1}f(a)a𝔬(S0){0,1}𝒩S0,f(a+y)𝑑yFS0,f(x)𝑑x.much-less-thansubscript𝑎superscript𝐹1𝑓𝑎subscript𝑎𝔬subscript𝑆001subscriptsubscript𝒩subscript𝑆0𝑓𝑎𝑦differential-d𝑦subscriptsubscript𝐹subscript𝑆0𝑓𝑥differential-d𝑥\displaystyle\sum_{a\in F^{\times}-\{1\}}f(a)\ll\sum_{a\in\mathfrak{o}(S_{0})-\{0,1\}}\int_{{\mathcal{N}}_{S_{0},\infty}}f(a+y)\,{{d}}y\leqslant\int_{F_{S_{0},\infty}}f(x)\,{{d}}x.

Since lv4subscript𝑙𝑣4l_{v}\geqslant 4 for all vΣ𝑣subscriptΣv\in\Sigma_{\infty}, the last integral is easily seen to be convergent. ∎

We have

𝕁hyp(𝐬,β)=subscript𝕁hyp𝐬𝛽absent\displaystyle{\mathbb{J}}_{\rm{hyp}}({\mathbf{s}},\beta)= 12aF×{1}𝔸𝑑x𝕂𝑑kΦ(𝐬;k1[a(a1)x01]k)12subscript𝑎superscript𝐹1subscript𝔸differential-d𝑥subscript𝕂differential-d𝑘Φ𝐬superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘\displaystyle\tfrac{1}{2}\sum_{a\in F^{\times}-\{1\}}\int_{{\mathbb{A}}}{{d}}x\int_{{\mathbb{K}}}{{d}}k\,\Phi\left({\mathbf{s}};k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)
×{Lσβ(z)ζF(z+12)ζF(z+12)vΣFφv(0,z)([1xv01])dz}absentsubscriptsubscript𝐿𝜎𝛽𝑧subscript𝜁𝐹𝑧12subscript𝜁𝐹𝑧12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣0𝑧delimited-[]1subscript𝑥𝑣01𝑑𝑧\displaystyle\times\left\{\int_{L_{\sigma}}\beta(z)\,\zeta_{F}\left(\tfrac{z+1}{2}\right)\zeta_{F}\left(\tfrac{-z+1}{2}\right)\,\prod_{v\in\Sigma_{F}}\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x_{v}\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}z\right\}

with 1<σ<11𝜎1-1<\sigma<1. By the estimate of Proposition 6.12, we can change the order of the integrals to have the formula

𝕁hyp(𝐬,β)subscript𝕁hyp𝐬𝛽\displaystyle{\mathbb{J}}_{\rm{hyp}}({\mathbf{s}},\beta) =12Lσβ(z)ζF(z+12)ζF(z+12){aF×{1}𝔉(z)(a)}𝑑z.absent12subscriptsubscript𝐿𝜎𝛽𝑧subscript𝜁𝐹𝑧12subscript𝜁𝐹𝑧12subscript𝑎superscript𝐹1superscript𝔉𝑧𝑎differential-d𝑧\displaystyle=\tfrac{1}{2}\,\int_{L_{\sigma}}\beta(z)\,\zeta_{F}\left(\tfrac{z+1}{2}\right)\zeta_{F}\left(\tfrac{-z+1}{2}\right)\,\{\sum_{a\in F^{\times}-\{1\}}{\mathfrak{F}}^{(z)}(a)\}\,{{d}}z.

Define

(6.13) J^hyp(𝐬,z)=12ζF(z+12)ζF(z+12){aF×{1}𝔉(z)(a)}.subscript^𝐽hyp𝐬𝑧12subscript𝜁𝐹𝑧12subscript𝜁𝐹𝑧12subscript𝑎superscript𝐹1superscript𝔉𝑧𝑎\displaystyle\hat{J}_{\rm hyp}({\mathbf{s}},z)=\tfrac{1}{2}\,\zeta_{F}\left(\tfrac{z+1}{2}\right)\zeta_{F}\left(\tfrac{-z+1}{2}\right)\,\{\sum_{a\in F^{\times}-\{1\}}{\mathfrak{F}}^{(z)}(a)\}.

Then we obtain a precise form of Theorem 4.1 for the F𝐹F-hyperbolic term.

Proposition 6.13.

Suppose Re(sv)>2Resubscript𝑠𝑣2\operatorname{Re}(s_{v})>2 for all vS𝑣𝑆v\in S. The function J^hyp(𝐬,z)subscript^𝐽hyp𝐬𝑧\hat{J}_{\rm hyp}({\mathbf{s}},z) is holomorphic away from z=±1𝑧plus-or-minus1z=\pm 1 and vertically of moderate growth on |Re(z)|<min({lv3|vΣ}{Re(sv)2|vS})Re𝑧conditional-setsubscript𝑙𝑣3𝑣subscriptΣconditional-setResubscript𝑠𝑣2𝑣𝑆|\operatorname{Re}(z)|<\min\,(\{l_{v}-3|\,v\in\Sigma_{\infty}\}\cup\{\operatorname{Re}(s_{v})-2|\,v\in S\,\}). For any contour Lσsubscript𝐿𝜎L_{\sigma} (|σ|<1)𝜎1(|\sigma|<1), we have the formula (4.7) with =hyphyp\natural={\rm hyp}.

7. The F𝐹F-elliptic term

In this section, we study 𝕁ell(𝐬;β)subscript𝕁ell𝐬𝛽{\mathbb{J}}_{\rm ell}({\mathbf{s}};\beta) to show Theorem 4.1 for the F𝐹F-elliptic term. From this section on, we suppose that 222 splits completely in the extension F/𝐹F/{\mathbb{Q}}; thus Fv2subscript𝐹𝑣subscript2F_{v}\cong{\mathbb{Q}}_{2} for all vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}.

7.1. Parametrization of elliptic elements

Set QF={(t,n)F2|t24n0}subscript𝑄𝐹conditional-set𝑡𝑛superscript𝐹2superscript𝑡24𝑛0Q_{F}=\{(t,n)\in F^{2}|\,t^{2}-4n\not=0\,\}. Let us say that two elements (t,n)𝑡𝑛(t,n) and (t,n)superscript𝑡superscript𝑛(t^{\prime},n^{\prime}) from QFsubscript𝑄𝐹Q_{F} are F𝐹F-equivalent if there exists cF×𝑐superscript𝐹c\in F^{\times} such that (t,n)=(ct,c2n)superscript𝑡superscript𝑛𝑐𝑡superscript𝑐2𝑛(t^{\prime},n^{\prime})=(ct,c^{2}n). The F𝐹F-equivalence class of a pair (t,n)QF𝑡𝑛subscript𝑄𝐹(t,n)\in Q_{F} is denoted by (t:n)F(t:n)_{F}. The quotient set of QFsubscript𝑄𝐹Q_{F} by the F𝐹F-equivalence relation is denoted by 𝒬Fsubscript𝒬𝐹{\mathcal{Q}}_{F}, i.e.,

𝒬F={(t:n)F|t,nF,t24n0}.{\mathcal{Q}}_{F}=\{(t:n)_{F}|\,t,\,n\in F,\,t^{2}-4n\not=0\}.

Let 𝒬FIrrsuperscriptsubscript𝒬𝐹Irr{\mathcal{Q}}_{F}^{\rm Irr} be the set of (t:n)F𝒬F(t:n)_{F}\in{\mathcal{Q}}_{F} such that the polynomial X2tX+nsuperscript𝑋2𝑡𝑋𝑛X^{2}-tX+n is F𝐹F-irreducible. For γ~𝒬FIrr~𝛾superscriptsubscript𝒬𝐹Irr\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm Irr}, fix its representative (t,n)QF𝑡𝑛subscript𝑄𝐹(t,n)\in Q_{F} once and for all and set γ=[t/21Δ/4t/2]GF𝛾delimited-[]𝑡21Δ4𝑡2subscript𝐺𝐹\gamma=\left[\begin{smallmatrix}t/2&1\\ \Delta/4&t/2\end{smallmatrix}\right]\in G_{F} and Δ=t24nΔsuperscript𝑡24𝑛\Delta=t^{2}-4n. The GFsubscript𝐺𝐹G_{F}-conjugacy class with characteristic polynomial t2tX+nsuperscript𝑡2𝑡𝑋𝑛t^{2}-tX+n is represented by the element γ𝛾\gamma. For a quadratic extension EF[X]/(X2Δ/4)𝐸𝐹delimited-[]𝑋superscript𝑋2Δ4E\cong F[X]/(X^{2}-\Delta/4) of F𝐹F with a prescribed square root ΔE×Δsuperscript𝐸\sqrt{\Delta}\in E^{\times}, let ιΔ:EM2(F):subscript𝜄Δ𝐸subscriptM2𝐹\iota_{\Delta}:E\hookrightarrow{\operatorname{M}}_{2}(F) be the F𝐹F-algebra embedding

ιΔ(a+bΔ/2)=[abΔb/4a],a,bF.formulae-sequencesubscript𝜄Δ𝑎𝑏Δ2delimited-[]𝑎𝑏Δ𝑏4𝑎𝑎𝑏𝐹\iota_{\Delta}(a+b\sqrt{\Delta}/2)=\left[\begin{smallmatrix}a&b\\ \Delta b/4&a\end{smallmatrix}\right],\quad a,b\in F.

Then the centralizer of γ𝛾\gamma in GFsubscript𝐺𝐹G_{F} is Gγ,F=ιΔ(E×)subscript𝐺𝛾𝐹subscript𝜄Δsuperscript𝐸G_{\gamma,F}=\iota_{\Delta}(E^{\times}).

For any place vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}, we can write the image of 41Δsuperscript41Δ4^{-1}\Delta in Fvsubscript𝐹𝑣F_{v} as 41Δ=Δv0mv2superscript41ΔsuperscriptsubscriptΔ𝑣0superscriptsubscript𝑚𝑣24^{-1}\Delta=\Delta_{v}^{0}\,m_{v}^{2} with mvFv×,Δv0Fv×/(Fv×)2formulae-sequencesubscript𝑚𝑣superscriptsubscript𝐹𝑣superscriptsubscriptΔ𝑣0superscriptsubscript𝐹𝑣superscriptsuperscriptsubscript𝐹𝑣2m_{v}\in F_{v}^{\times},\,\Delta_{v}^{0}\in F_{v}^{\times}/(F_{v}^{\times})^{2}; we suppose (a) vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}, Δv0superscriptsubscriptΔ𝑣0\Delta_{v}^{0} belongs to (𝔭v𝔭v2){1}(𝔬v×(𝔬v×)2)subscript𝔭𝑣superscriptsubscript𝔭𝑣21superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2({\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2})\cup\{1\}\cup(\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}), or (b) vΣ𝑣subscriptΣv\in\Sigma_{\infty} and Δv0{+1,1}superscriptsubscriptΔ𝑣011\Delta_{v}^{0}\in\{+1,-1\}. We fix such a factorization of Δ/4Δ4\Delta/4 in Fv×superscriptsubscript𝐹𝑣F_{v}^{\times}. Since 222 is assumed to be completely split in F/𝐹F/{\mathbb{Q}}, we have 𝔬v×/(𝔬v×)22×/(1+82)={±5,±1}superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2superscriptsubscript218subscript2plus-or-minus5plus-or-minus1\mathfrak{o}_{v}^{\times}/(\mathfrak{o}_{v}^{\times})^{2}\cong{\mathbb{Z}}_{2}^{\times}/(1+8{\mathbb{Z}}_{2})=\{\pm 5,\pm 1\} thus may suppose Δv0{±5,±1,±10,±2}superscriptsubscriptΔ𝑣0plus-or-minus5plus-or-minus1plus-or-minus10plus-or-minus2\Delta_{v}^{0}\in\{\pm 5,\pm 1,\pm 10,\pm 2\} for all vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}. Let 𝔡E/Fsubscript𝔡𝐸𝐹{\mathfrak{d}}_{E/F} denote the relative discriminant of E/F𝐸𝐹E/F. From this, it is easily seen that 𝔡E/F𝔬v=Δv0𝔬vsubscript𝔡𝐸𝐹subscript𝔬𝑣superscriptsubscriptΔ𝑣0subscript𝔬𝑣{\mathfrak{d}}_{E/F}\mathfrak{o}_{v}=\Delta_{v}^{0}\mathfrak{o}_{v} if vΣfinΣdyadic𝑣subscriptΣfinsubscriptΣdyadicv\in\Sigma_{\rm fin}-\Sigma_{\rm dyadic} or vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}, Δv0{5,1}superscriptsubscriptΔ𝑣051\Delta_{v}^{0}\in\{5,1\}, and 𝔡E/F𝔬v=4Δv0𝔬vsubscript𝔡𝐸𝐹subscript𝔬𝑣4superscriptsubscriptΔ𝑣0subscript𝔬𝑣{\mathfrak{d}}_{E/F}\mathfrak{o}_{v}=4\Delta_{v}^{0}\mathfrak{o}_{v} for vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}, Δv0{5,1,±10,±2}superscriptsubscriptΔ𝑣051plus-or-minus10plus-or-minus2\Delta_{v}^{0}\in\{-5,-1,\pm 10,\pm 2\}. Let εΔsubscript𝜀Δ\varepsilon_{\Delta} be idele class character associated with the quadratic extension E=F(Δ)𝐸𝐹ΔE=F(\sqrt{\Delta}) by class field theory. Then ζE(z)=L(z,εΔ)ζF(z)subscript𝜁𝐸𝑧𝐿𝑧subscript𝜀Δsubscript𝜁𝐹𝑧\zeta_{E}(z)=L(z,\varepsilon_{\Delta})\zeta_{F}(z) and the conductor of εΔsubscript𝜀Δ\varepsilon_{\Delta} is 𝔡E/Fsubscript𝔡𝐸𝐹{\mathfrak{d}}_{E/F}.

We have the direct sum decomposition Ev=Fv+Δv0Fvsubscript𝐸𝑣subscript𝐹𝑣superscriptsubscriptΔ𝑣0subscript𝐹𝑣E_{v}=F_{v}+\sqrt{\Delta_{v}^{0}}F_{v} as Fvsubscript𝐹𝑣F_{v}-vector spaces, which determines an Fvsubscript𝐹𝑣F_{v}-embedding ιΔv0:EvM2(Fv):subscript𝜄superscriptsubscriptΔ𝑣0subscript𝐸𝑣subscriptM2subscript𝐹𝑣\iota_{\Delta_{v}^{0}}:E_{v}\hookrightarrow{\operatorname{M}}_{2}(F_{v}) as

ιΔv0(a+bΔv0)=[abbΔv0a],a,bFv if Δv01,formulae-sequencesubscript𝜄superscriptsubscriptΔ𝑣0𝑎𝑏superscriptsubscriptΔ𝑣0delimited-[]𝑎𝑏𝑏superscriptsubscriptΔ𝑣0𝑎𝑎𝑏subscript𝐹𝑣 if Δv01,\displaystyle\iota_{\Delta_{v}^{0}}(a+b\sqrt{\Delta_{v}^{0}})=\left[\begin{smallmatrix}a&b\\ b\Delta_{v}^{0}&a\end{smallmatrix}\right],\quad a,b\in F_{v}\quad\text{ if $\Delta_{v}^{0}\neq 1$,}
ιΔv0(a+bΔv0)=[a+b00ab],a,bFv if Δv0=1.formulae-sequencesubscript𝜄superscriptsubscriptΔ𝑣0𝑎𝑏superscriptsubscriptΔ𝑣0delimited-[]𝑎𝑏00𝑎𝑏𝑎𝑏subscript𝐹𝑣 if Δv0=1.\displaystyle\iota_{\Delta_{v}^{0}}(a+b\sqrt{\Delta_{v}^{0}})=\left[\begin{smallmatrix}a+b&0\\ 0&a-b\end{smallmatrix}\right],\quad a,b\in F_{v}\quad\text{ if $\Delta_{v}^{0}=1$.}

Set 𝔗Δ,v=ιΔv0(Ev×)subscript𝔗Δ𝑣subscript𝜄superscriptsubscriptΔ𝑣0superscriptsubscript𝐸𝑣{\mathfrak{T}}_{\Delta,v}=\iota_{\Delta_{v}^{0}}(E_{v}^{\times}). For vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}, we set 𝔗Δ,v+=Zv𝕂v𝔗Δ,vsuperscriptsubscript𝔗Δ𝑣subscript𝑍𝑣subscript𝕂𝑣subscript𝔗Δ𝑣{\mathfrak{T}}_{\Delta,v}^{+}=Z_{v}{\mathbb{K}}_{v}\cap{\mathfrak{T}}_{\Delta,v}.

Lemma 7.1.

Let vΣfinΣdyadic𝑣subscriptΣfinsubscriptΣdyadicv\in\Sigma_{\rm fin}-\Sigma_{\rm dyadic}. If τ𝔬v×(𝔬v×)2𝜏superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\tau\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}, then a2τ𝔬v×superscript𝑎2𝜏superscriptsubscript𝔬𝑣a^{2}-\tau\in\mathfrak{o}_{v}^{\times} for all a𝔬v𝑎subscript𝔬𝑣a\in\mathfrak{o}_{v}.

Proof.

If a2τ𝔭vsuperscript𝑎2𝜏subscript𝔭𝑣a^{2}-\tau\in{\mathfrak{p}}_{v}, then a𝔬v×𝑎superscriptsubscript𝔬𝑣a\in\mathfrak{o}_{v}^{\times} and τa2(mod𝔭v)𝜏annotatedsuperscript𝑎2pmodsubscript𝔭𝑣\tau\equiv a^{2}\pmod{{\mathfrak{p}}_{v}}. Since τ(𝔬v×)2𝜏superscriptsuperscriptsubscript𝔬𝑣2\tau\not\in(\mathfrak{o}_{v}^{\times})^{2}, this is impossible by Hensel’s lemma. ∎

Lemma 7.2.

Let vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}.

  • (i)

    If vΣfinΣdyadic𝑣subscriptΣfinsubscriptΣdyadicv\in\Sigma_{\rm fin}-\Sigma_{\rm dyadic} and Δv0𝔬v×(𝔬v×)2superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}, then 𝔗Δ,v+=𝔗Δ,vsuperscriptsubscript𝔗Δ𝑣subscript𝔗Δ𝑣{\mathfrak{T}}_{\Delta,v}^{+}={\mathfrak{T}}_{\Delta,v}.

  • (ii)

    If vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and Δv0𝔭v𝔭v2superscriptsubscriptΔ𝑣0subscript𝔭𝑣superscriptsubscript𝔭𝑣2\Delta_{v}^{0}\in{\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2}, then 𝔗Δ,v=𝔗Δ,v+[01Δv00]𝔗Δ,v+subscript𝔗Δ𝑣superscriptsubscript𝔗Δ𝑣delimited-[]01superscriptsubscriptΔ𝑣00superscriptsubscript𝔗Δ𝑣{\mathfrak{T}}_{\Delta,v}={\mathfrak{T}}_{\Delta,v}^{+}\cup\left[\begin{smallmatrix}0&1\\ \Delta_{v}^{0}&0\end{smallmatrix}\right]{\mathfrak{T}}_{\Delta,v}^{+}.

  • (iii)

    If vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} and Δv0{1,5}superscriptsubscriptΔ𝑣015\Delta_{v}^{0}\in\{-1,-5\}, then 𝔗Δ,v=𝔗Δ,v+[11Δv01]𝔗Δ,v+subscript𝔗Δ𝑣superscriptsubscript𝔗Δ𝑣delimited-[]11superscriptsubscriptΔ𝑣01superscriptsubscript𝔗Δ𝑣{\mathfrak{T}}_{\Delta,v}={\mathfrak{T}}_{\Delta,v}^{+}\cup\left[\begin{smallmatrix}1&1\\ \Delta_{v}^{0}&1\end{smallmatrix}\right]{\mathfrak{T}}_{\Delta,v}^{+}.

  • (iv)

    If vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} and Δv0=5superscriptsubscriptΔ𝑣05\Delta_{v}^{0}=5, then 𝔗Δ,v=𝔗Δ,v+[1151]𝔗Δ,v+[3153]𝔗Δ,v+.subscript𝔗Δ𝑣square-unionsuperscriptsubscript𝔗Δ𝑣delimited-[]1151superscriptsubscript𝔗Δ𝑣delimited-[]3153superscriptsubscript𝔗Δ𝑣{\mathfrak{T}}_{\Delta,v}={\mathfrak{T}}_{\Delta,v}^{+}\sqcup[\begin{smallmatrix}1&1\\ 5&1\end{smallmatrix}]{\mathfrak{T}}_{\Delta,v}^{+}\sqcup[\begin{smallmatrix}3&1\\ 5&3\end{smallmatrix}]{\mathfrak{T}}_{\Delta,v}^{+}.

  • (v)

    If vΣ𝑣subscriptΣv\in\Sigma_{\infty}, Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=-1, then 𝔗Δ,v=Zv𝕂v0subscript𝔗Δ𝑣subscript𝑍𝑣superscriptsubscript𝕂𝑣0{\mathfrak{T}}_{\Delta,v}=Z_{v}{\mathbb{K}}_{v}^{0}.

Proof.

(i) For (x,y)Fv2{(0,0)}𝑥𝑦superscriptsubscript𝐹𝑣200(x,y)\in F_{v}^{2}-\{(0,0)\}, we have [xyΔv0yx]=[x00x][1y/xΔv0y/x1]delimited-[]𝑥𝑦superscriptsubscriptΔ𝑣0𝑦𝑥delimited-[]𝑥00𝑥delimited-[]1𝑦𝑥superscriptsubscriptΔ𝑣0𝑦𝑥1[\begin{smallmatrix}x&y\\ \Delta_{v}^{0}y&x\end{smallmatrix}]=[\begin{smallmatrix}x&0\\ 0&x\end{smallmatrix}][\begin{smallmatrix}1&y/x\\ \Delta_{v}^{0}y/x&1\end{smallmatrix}]. Then, 1Δv0y2/x2𝔬v×1superscriptsubscriptΔ𝑣0superscript𝑦2superscript𝑥2superscriptsubscript𝔬𝑣1-\Delta_{v}^{0}y^{2}/x^{2}\in{\mathfrak{o}}_{v}^{\times} if |x|v>|y|vsubscript𝑥𝑣subscript𝑦𝑣|x|_{v}>|y|_{v}. If |x|v=|y|vsubscript𝑥𝑣subscript𝑦𝑣|x|_{v}=|y|_{v}, by putting x=ϖvmux𝑥superscriptsubscriptitalic-ϖ𝑣𝑚subscript𝑢𝑥x=\varpi_{v}^{m}u_{x} and y=ϖvmuy𝑦superscriptsubscriptitalic-ϖ𝑣𝑚subscript𝑢𝑦y=\varpi_{v}^{m}u_{y} with ux,uy𝔬v×subscript𝑢𝑥subscript𝑢𝑦superscriptsubscript𝔬𝑣u_{x},u_{y}\in{\mathfrak{o}}_{v}^{\times} and m𝑚m\in\mathbb{Z}, we have [xΔv0yyx]=[ϖvm00ϖvm][uxuyΔv0uyux]delimited-[]𝑥superscriptsubscriptΔ𝑣0𝑦𝑦𝑥delimited-[]superscriptsubscriptitalic-ϖ𝑣𝑚00superscriptsubscriptitalic-ϖ𝑣𝑚delimited-[]subscript𝑢𝑥subscript𝑢𝑦superscriptsubscriptΔ𝑣0subscript𝑢𝑦subscript𝑢𝑥[\begin{smallmatrix}x&\Delta_{v}^{0}y\\ y&x\end{smallmatrix}]=[\begin{smallmatrix}\varpi_{v}^{m}&0\\ 0&\varpi_{v}^{m}\end{smallmatrix}][\begin{smallmatrix}u_{x}&u_{y}\\ \Delta_{v}^{0}u_{y}&u_{x}\end{smallmatrix}], and ux2Δv0uy2𝔬v×superscriptsubscript𝑢𝑥2superscriptsubscriptΔ𝑣0superscriptsubscript𝑢𝑦2superscriptsubscript𝔬𝑣u_{x}^{2}-\Delta_{v}^{0}\,u_{y}^{2}\in\mathfrak{o}_{v}^{\times} by Lemmma 7.1. Thus we are done. (ii) and (iii) follow from the observation that an element ιΔv0(a+bΔv0)subscript𝜄superscriptsubscriptΔ𝑣0𝑎𝑏superscriptsubscriptΔ𝑣0\iota_{\Delta_{v}^{0}}(a+b\sqrt{\Delta_{v}^{0}}) belongs to 𝔗Δ,v+superscriptsubscript𝔗Δ𝑣{\mathfrak{T}}_{\Delta,v}^{+} if and only if ordv(a2b2Δv0)2subscriptord𝑣superscript𝑎2superscript𝑏2superscriptsubscriptΔ𝑣02{\rm{ord}}_{v}(a^{2}-b^{2}\Delta_{v}^{0})\in 2{\mathbb{Z}}. (iv) Set Mj={[u1Δv0u]M2(2)|uj+42}subscript𝑀𝑗conditional-setdelimited-[]𝑢1superscriptsubscriptΔ𝑣0𝑢subscriptM2subscript2𝑢𝑗4subscript2M_{j}=\{[\begin{smallmatrix}u&1\\ \Delta_{v}^{0}&u\end{smallmatrix}]\in{\rm M}_{2}(\mathbb{Z}_{2})\ |\ u\in j+4\mathbb{Z}_{2}\} for j=1,3𝑗13j=1,3. Then ZvMvsubscript𝑍𝑣subscript𝑀𝑣Z_{v}M_{v} with Mv=𝕂v𝔗Δ,vsubscript𝑀𝑣subscript𝕂𝑣subscript𝔗Δ𝑣M_{v}={\mathbb{K}}_{v}\cap{\mathfrak{T}}_{\Delta,v} is written as ZvMv=ZvM1ZvM3=[1151]𝔗Δ,v+[3153]𝔗Δ,v+.subscript𝑍𝑣subscript𝑀𝑣square-unionsubscript𝑍𝑣subscript𝑀1subscript𝑍𝑣subscript𝑀3square-uniondelimited-[]1151superscriptsubscript𝔗Δ𝑣delimited-[]3153superscriptsubscript𝔗Δ𝑣Z_{v}M_{v}=Z_{v}M_{1}\sqcup Z_{v}M_{3}=[\begin{smallmatrix}1&1\\ 5&1\end{smallmatrix}]{\mathfrak{T}}_{\Delta,v}^{+}\sqcup[\begin{smallmatrix}3&1\\ 5&3\end{smallmatrix}]{\mathfrak{T}}_{\Delta,v}^{+}. By 𝔗Δ,v=𝔗Δ,v+ZvMvsubscript𝔗Δ𝑣square-unionsuperscriptsubscript𝔗Δ𝑣subscript𝑍𝑣subscript𝑀𝑣{\mathfrak{T}}_{\Delta,v}={\mathfrak{T}}_{\Delta,v}^{+}\sqcup Z_{v}M_{v}, we are done. (v) is confirmed by direct computation. ∎

For vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}, set

RΔ,v=[mv001]if Δv01, andRΔ,v=[1111][mv001]if Δv0=1.formulae-sequencesubscript𝑅Δ𝑣delimited-[]subscript𝑚𝑣001if Δv01, andsubscript𝑅Δ𝑣delimited-[]1111delimited-[]subscript𝑚𝑣001if Δv0=1R_{\Delta,v}=\left[\begin{smallmatrix}m_{v}&0\\ 0&1\end{smallmatrix}\right]\quad\text{if $\Delta_{v}^{0}\not=1$, and}\,\,R_{\Delta,v}=\left[\begin{smallmatrix}1&1\\ 1&-1\end{smallmatrix}\right]\left[\begin{smallmatrix}m_{v}&0\\ 0&1\end{smallmatrix}\right]\quad\text{if $\Delta_{v}^{0}=1$}.

Then we have the relation

(7.1) RΔ,v1𝔗Δ,vRΔ,v=Gγ,v.superscriptsubscript𝑅Δ𝑣1subscript𝔗Δ𝑣subscript𝑅Δ𝑣subscript𝐺𝛾𝑣\displaystyle R_{\Delta,v}^{-1}{\mathfrak{T}}_{\Delta,v}R_{\Delta,v}=G_{\gamma,v}.

Since Δ𝔬v×Δsuperscriptsubscript𝔬𝑣\Delta\in\mathfrak{o}_{v}^{\times} for almost all v𝑣v, the system RΔ={RΔ,v}vΣFsubscript𝑅Δsubscriptsubscript𝑅Δ𝑣𝑣subscriptΣ𝐹R_{\Delta}=\{R_{\Delta,v}\}_{v\in\Sigma_{F}} belongs to G𝔸subscript𝐺𝔸G_{\mathbb{A}}. Define

𝔗Δ={(hv)vΣF𝔗Δ,v|hv𝕂v𝔗Δ,v for almost all vΣfin}.subscript𝔗Δconditional-setsubscript𝑣subscriptproduct𝑣subscriptΣ𝐹subscript𝔗Δ𝑣hv𝕂v𝔗Δ,v for almost all vΣfin\displaystyle{\mathfrak{T}}_{\Delta}=\{(h_{v})\in\prod_{v\in\Sigma_{F}}{\mathfrak{T}}_{\Delta,v}|\,\text{$h_{v}\in{\mathbb{K}}_{v}\cap{\mathfrak{T}}_{\Delta,v}$ for almost all $v\in\Sigma_{\rm fin}$}\,\}.

If we view 𝔗Δsubscript𝔗Δ{\mathfrak{T}}_{\Delta} as a closed subgroup of G𝔸subscript𝐺𝔸G_{\mathbb{A}}, then

(7.2) RΔ1𝔗ΔRΔ=Gγ,𝔸.superscriptsubscript𝑅Δ1subscript𝔗Δsubscript𝑅Δsubscript𝐺𝛾𝔸\displaystyle R_{\Delta}^{-1}\,{\mathfrak{T}}_{\Delta}R_{\Delta}=G_{\gamma,{\mathbb{A}}}.

For each place v𝑣v, we fix a Haar measure on Ev×superscriptsubscript𝐸𝑣E_{v}^{\times} by

d×τv=ζEv(1)|xv241Δyv2|v1dxvdyvwith τv=xv+21Δyv(xv,yvFv),superscript𝑑subscript𝜏𝑣subscript𝜁subscript𝐸𝑣1superscriptsubscriptsuperscriptsubscript𝑥𝑣2superscript41Δsuperscriptsubscript𝑦𝑣2𝑣1𝑑subscript𝑥𝑣𝑑subscript𝑦𝑣with τv=xv+21Δyv(xv,yvFv),\displaystyle{{d}}^{\times}\tau_{v}=\zeta_{E_{v}}(1){|x_{v}^{2}-4^{-1}\Delta y_{v}^{2}|_{v}^{-1}}{{d}}x_{v}\,{{d}}y_{v}\quad\text{with $\tau_{v}=x_{v}+2^{-1}\sqrt{\Delta}y_{v}\,(x_{v},y_{v}\in F_{v})$,}

and transfer this to Gγ,vsubscript𝐺𝛾𝑣G_{\gamma,v} by ιΔ:Ev×Gγ,v:subscript𝜄Δsuperscriptsubscript𝐸𝑣subscript𝐺𝛾𝑣\iota_{\Delta}:E_{v}^{\times}\cong G_{\gamma,v}. We transfer the Haar measure d×τ=vd×τv{{d}}^{\times}\tau=\otimes_{v}{{d}}^{\times}\tau_{v} on 𝔸E×superscriptsubscript𝔸𝐸{\mathbb{A}}_{E}^{\times} to Gγ,𝔸subscript𝐺𝛾𝔸G_{\gamma,{\mathbb{A}}} to define a Haar measure on Gγ,𝔸subscript𝐺𝛾𝔸G_{\gamma,{\mathbb{A}}}. We use the relations (7.1) and (7.2) to define Haar measures on groups 𝔗Δ,vsubscript𝔗Δ𝑣{\mathfrak{T}}_{\Delta,v} and 𝔗Δsubscript𝔗Δ{\mathfrak{T}}_{\Delta}.

Lemma 7.3.

If vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}, then vol(Zv\𝔗Δ,v+)=|mv|v1qvdv/2vol\subscript𝑍𝑣superscriptsubscript𝔗Δ𝑣superscriptsubscriptsubscript𝑚𝑣𝑣1subscriptsuperscript𝑞subscript𝑑𝑣2𝑣{\operatorname{vol}}(Z_{v}\backslash{\mathfrak{T}}_{\Delta,v}^{+})=|m_{v}|_{v}^{-1}q^{-d_{v}/2}_{v} unless vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}, Δv0=5superscriptsubscriptΔ𝑣05\Delta_{v}^{0}=5, in which case vol(Zv\𝔗Δ,v+)=23|mv|v12dv/2.vol\subscript𝑍𝑣superscriptsubscript𝔗Δ𝑣23superscriptsubscriptsubscript𝑚𝑣𝑣1superscript2subscript𝑑𝑣2{\operatorname{vol}}(Z_{v}\backslash{\mathfrak{T}}_{\Delta,v}^{+})=\tfrac{2}{3}\,|m_{v}|_{v}^{-1}2^{-d_{v}/2}. If vΣ𝑣subscriptΣv\in\Sigma_{\infty}, then vol(Zv\𝔗Δ,v)=|mv|v1vol\subscript𝑍𝑣subscript𝔗Δ𝑣superscriptsubscriptsubscript𝑚𝑣𝑣1{\operatorname{vol}}(Z_{v}\backslash{\mathfrak{T}}_{\Delta,v})=|m_{v}|_{v}^{-1}.

Proof.

Let vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. Then, 𝔗Δ,v+=ZvMvsuperscriptsubscript𝔗Δ𝑣subscript𝑍𝑣subscript𝑀𝑣\mathfrak{T}_{\Delta,v}^{+}=Z_{v}M_{v} with Mv=𝕂v𝔗Δ,vsubscript𝑀𝑣subscript𝕂𝑣subscript𝔗Δ𝑣M_{v}={\mathbb{K}}_{v}\cap{\mathfrak{T}}_{\Delta,v}, and chZvMv(t)=vol(ZvMv)1zZvchMv(zt)dz{\rm ch}_{Z_{v}M_{v}}(t)={\operatorname{vol}}(Z_{v}\cap M_{v})^{-1}\int_{z\in Z_{v}}{\rm ch}_{M_{v}}(zt)\,{{d}}z, as in the proof of [14, Lemma 7.39]. We have

vol(Zv\𝔗Δ,v+)=vol(ZvMv)1𝔗Δ,vchMv(τ)d×τ=qvdv/2a,b𝔬va2Δv0mv2b2𝔬v×ζEv(1)dadb|a2Δv0(mvb)2|v.\displaystyle{\operatorname{vol}}(Z_{v}\backslash\mathfrak{T}_{\Delta,v}^{+})={\operatorname{vol}}(Z_{v}\cap M_{v})^{-1}\textstyle{\int}_{{\mathfrak{T}}_{\Delta,v}}{\rm ch}_{M_{v}}(\tau)\,{{d}}^{\times}\tau=q_{v}^{d_{v}/2}{\textstyle\iint}_{\begin{subarray}{c}a,b\in\mathfrak{o}_{v}\\ a^{2}-\Delta_{v}^{0}m_{v}^{2}b^{2}\in\mathfrak{o}_{v}^{\times}\end{subarray}}\zeta_{E_{v}}(1)\tfrac{{{d}}a{{d}}b}{|a^{2}-\Delta^{0}_{v}(m_{v}b)^{2}|_{v}}.

This equals |mv|v1qvdv/2superscriptsubscriptsubscript𝑚𝑣𝑣1superscriptsubscript𝑞𝑣subscript𝑑𝑣2|m_{v}|_{v}^{-1}q_{v}^{-d_{v}/2} unless |2|v<1subscript2𝑣1|2|_{v}<1 and τ=5𝜏5\tau=5. If |2|v<1subscript2𝑣1|2|_{v}<1 and τ=5𝜏5\tau=5, then Ev=2(5)subscript𝐸𝑣subscript25E_{v}={\mathbb{Q}}_{2}(\sqrt{5}) and ζEv(1)=(141)1=4/3subscript𝜁subscript𝐸𝑣1superscript1superscript41143\zeta_{E_{v}}(1)=(1-4^{-1})^{-1}=4/3; thus vol(Zv\𝔗Δ,v+)=(4/3)qvdv/2|mv|v1U𝑑a𝑑bvol\subscript𝑍𝑣superscriptsubscript𝔗Δ𝑣43superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscriptsubscript𝑚𝑣𝑣1subscriptdouble-integral𝑈differential-d𝑎differential-d𝑏{\operatorname{vol}}(Z_{v}\backslash{\mathfrak{T}}_{\Delta,v}^{+})=(4/3)q_{v}^{d_{v}/2}|m_{v}|_{v}^{-1}\,\iint_{U}{{d}}a{{d}}b with U={(a,b)2|a25b22×}𝑈conditional-set𝑎𝑏subscript2superscript𝑎25superscript𝑏2superscriptsubscript2U=\{(a,b)\in{\mathbb{Z}}_{2}|\,a^{2}-5b^{2}\in{\mathbb{Z}}_{2}^{\times}\}. Since U=2×2{(22)×(22)2××2×}𝑈subscript2subscript2square-union2subscript22subscript2superscriptsubscript2subscriptsuperscript2U={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-\{(2{\mathbb{Z}}_{2})\times(2{\mathbb{Z}}_{2})\sqcup{\mathbb{Z}}_{2}^{\times}\times{\mathbb{Z}}^{\times}_{2}\}, we have vol(U)=1(1/2)2(1/2)2=1/2vol𝑈1superscript122superscript12212{\operatorname{vol}}(U)=1-(1/2)^{2}-(1/2)^{2}=1/2. Hence vol(Zv\𝔗Δ,v+)=qvdv/2|mv|v12/3vol\subscript𝑍𝑣superscriptsubscript𝔗Δ𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscriptsubscriptsubscript𝑚𝑣𝑣123{\operatorname{vol}}(Z_{v}\backslash{\mathfrak{T}}_{\Delta,v}^{+})=q_{v}^{-d_{v}/2}|m_{v}|_{v}^{-1}2/3 as desired. Let vΣ𝑣subscriptΣv\in\Sigma_{\infty} and Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=-1. Then 𝔗Δ,v={[abba]|a,b,a2+b20}subscript𝔗Δ𝑣conditional-setdelimited-[]𝑎𝑏𝑏𝑎formulae-sequence𝑎𝑏superscript𝑎2superscript𝑏20{\mathfrak{T}}_{\Delta,v}=\{\left[\begin{smallmatrix}a&-b\\ b&a\end{smallmatrix}\right]|\,a,b\in{\mathbb{R}},\,a^{2}+b^{2}\not=0\} is a direct product of +12subscriptsubscript12{\mathbb{R}}_{+}1_{2} and SO(2)SO2{\operatorname{SO}}(2) and d×τ=|mv|v1Γ(1)drrdθsuperscript𝑑𝜏superscriptsubscriptsubscript𝑚𝑣𝑣1subscriptΓ1𝑑𝑟𝑟𝑑𝜃{{d}}^{\times}\tau=|m_{v}|_{v}^{-1}\Gamma_{\mathbb{C}}(1)\tfrac{{{d}}r}{r}\,{{d}}\theta, where dr𝑑𝑟{{d}}r is the Lebesgue measure on {\mathbb{R}} and SO(2)𝑑θ=2πsubscriptSO2differential-d𝜃2𝜋\int_{{\operatorname{SO}}(2)}{{d}}\theta=2\pi. Thus vol(Zv\𝔗Δ,v)=21vol(+12\𝔗Δ,v)=|mv|v1vol\subscript𝑍𝑣subscript𝔗Δ𝑣superscript21vol\subscriptsubscript12subscript𝔗Δ𝑣superscriptsubscriptsubscript𝑚𝑣𝑣1{\operatorname{vol}}(Z_{v}\backslash{\mathfrak{T}}_{\Delta,v})=2^{-1}{\operatorname{vol}}(\mathbb{R}_{+}1_{2}\backslash{\mathfrak{T}}_{\Delta,v})=|m_{v}|_{v}^{-1}. ∎

Using the relation (7.2), we compute

𝕁ell(𝐬,β)subscript𝕁ell𝐬𝛽\displaystyle{\mathbb{J}}_{\rm ell}({\mathbf{s}},\beta) =12γ~𝒬FIrrZ𝔸GF\G𝔸ξGγ,F\GFΦ(𝐬;g1ξ1γξg)β(g)dgabsent12subscript~𝛾superscriptsubscript𝒬𝐹Irrsubscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸subscript𝜉\subscript𝐺𝛾𝐹subscript𝐺𝐹Φ𝐬superscript𝑔1superscript𝜉1𝛾𝜉𝑔superscriptsubscript𝛽𝑔𝑑𝑔\displaystyle=\tfrac{1}{2}\sum_{\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm Irr}}\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}\sum_{\xi\in G_{\gamma,F}\backslash G_{F}}\Phi({\mathbf{s}};g^{-1}\xi^{-1}\gamma\xi g){\mathcal{E}}_{\beta}^{*}(g)dg
=12γ~𝒬FIrrGγ,𝔸\G𝔸Φ(𝐬;g1γg){Z𝔸Gγ,F\Gγ,𝔸β(hg)𝑑h}𝑑gabsent12subscript~𝛾superscriptsubscript𝒬𝐹Irrsubscript\subscript𝐺𝛾𝔸subscript𝐺𝔸Φ𝐬superscript𝑔1𝛾𝑔subscript\subscript𝑍𝔸subscript𝐺𝛾𝐹subscript𝐺𝛾𝔸subscriptsuperscript𝛽𝑔differential-ddifferential-d𝑔\displaystyle=\tfrac{1}{2}\sum_{\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm Irr}}\int_{G_{\gamma,{\mathbb{A}}}\backslash G_{\mathbb{A}}}\Phi({\mathbf{s}};g^{-1}\gamma g)\,\{\int_{Z_{\mathbb{A}}G_{\gamma,F}\backslash G_{\gamma,{\mathbb{A}}}}{\mathcal{E}}^{*}_{\beta}(hg)\,{{d}}h\}\,{{d}}g
=12γ~𝒬FIrr𝔗Δ\G𝔸Φ(𝐬;g1RΔγRΔ1g)(β)Δ(g)𝑑gabsent12subscript~𝛾superscriptsubscript𝒬𝐹Irrsubscript\subscript𝔗Δsubscript𝐺𝔸Φ𝐬superscript𝑔1subscript𝑅Δ𝛾superscriptsubscript𝑅Δ1𝑔superscriptsuperscriptsubscript𝛽Δ𝑔differential-d𝑔\displaystyle=\tfrac{1}{2}\sum_{\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm Irr}}\int_{{\mathfrak{T}}_{\Delta}\backslash G_{\mathbb{A}}}\Phi({\mathbf{s}};g^{-1}R_{\Delta}\gamma R_{\Delta}^{-1}g)\,({\mathcal{E}}_{\beta}^{*})^{\Delta}(g)\,{{d}}g
(7.3) =12γ~𝒬FIrr𝔗Δ\G𝔸Φ(𝐬;g1γ^g)(β)Δ(g)𝑑g,absent12subscript~𝛾superscriptsubscript𝒬𝐹Irrsubscript\subscript𝔗Δsubscript𝐺𝔸Φ𝐬superscript𝑔1^𝛾𝑔superscriptsuperscriptsubscript𝛽Δ𝑔differential-d𝑔\displaystyle=\tfrac{1}{2}\sum_{\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm Irr}}\int_{{\mathfrak{T}}_{\Delta}\backslash G_{\mathbb{A}}}\Phi({\mathbf{s}};g^{-1}\hat{\gamma}g)\,({\mathcal{E}}_{\beta}^{*})^{\Delta}(g)\,{{d}}g,

where γ^=(γ^v)vΣF^𝛾subscriptsubscript^𝛾𝑣𝑣subscriptΣ𝐹\hat{\gamma}=(\hat{\gamma}_{v})_{v\in\Sigma_{F}} denotes the element of G𝔸subscript𝐺𝔸G_{\mathbb{A}} such that γ^v=[t2mv1Δv0t2mv]subscript^𝛾𝑣delimited-[]𝑡2subscript𝑚𝑣1superscriptsubscriptΔ𝑣0𝑡2subscript𝑚𝑣\hat{\gamma}_{v}=\left[\begin{smallmatrix}\frac{t}{2m_{v}}&1\\ \Delta_{v}^{0}&\frac{t}{2m_{v}}\end{smallmatrix}\right] if Δv01superscriptsubscriptΔ𝑣01\Delta_{v}^{0}\not=1 and γ^v=[t2mv+100t2mv1]subscript^𝛾𝑣delimited-[]𝑡2subscript𝑚𝑣100𝑡2subscript𝑚𝑣1\hat{\gamma}_{v}=\left[\begin{smallmatrix}\frac{t}{2m_{v}}+1&0\\ 0&\frac{t}{2m_{v}}-1\end{smallmatrix}\right] if Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1, and

(7.4) (β)Δ(g)superscriptsuperscriptsubscript𝛽Δ𝑔\displaystyle({\mathcal{E}}_{\beta}^{*})^{\Delta}(g) =Z𝔸Gγ,F\Gγ,𝔸β(hRΔ1g)𝑑h=𝔸×E×\𝔸E×β(ιΔ(τ)RΔ1g)d×τ.absentsubscript\subscript𝑍𝔸subscript𝐺𝛾𝐹subscript𝐺𝛾𝔸superscriptsubscript𝛽superscriptsubscript𝑅Δ1𝑔differential-dsubscript\superscript𝔸superscript𝐸superscriptsubscript𝔸𝐸superscriptsubscript𝛽subscript𝜄Δ𝜏superscriptsubscript𝑅Δ1𝑔superscript𝑑𝜏\displaystyle=\int_{Z_{\mathbb{A}}G_{\gamma,F}\backslash G_{\gamma,{\mathbb{A}}}}{\mathcal{E}}_{\beta}^{*}\left(hR_{\Delta}^{-1}g\right)\,{{d}}h=\int_{{\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times}}{\mathcal{E}}_{\beta}^{*}\left(\iota_{\Delta}(\tau)\,R_{\Delta}^{-1}g\right)\,{{d}}^{\times}\tau.

7.2. Periods of Eisenstein series along elliptic tori

We shall calculate the integral (7.4), which is absolutely convergent due to the compactness of 𝔸×E×\𝔸E×\superscript𝔸superscript𝐸superscriptsubscript𝔸𝐸{\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times}. To attain this, let us recall the multiplicity one property of the Waldspurger model of the principal series I(||vz/2)(z)I(|\,|_{v}^{z/2})\,(z\in{\mathbb{C}}) and the explicit formula of associated spherical function.

Lemma 7.4.

Suppose Δv01superscriptsubscriptΔ𝑣01\Delta_{v}^{0}\not=1. Let f0,vI(||vz/2)f_{0,v}\in I(|\,|_{v}^{z/2}) be a 𝕂vsubscript𝕂𝑣{\mathbb{K}}_{v}-invariant vector determined by f0,v(12)=1subscript𝑓0𝑣subscript121f_{0,v}(1_{2})=1, and set

φ0,v(g)=Zv\𝔗Δ,vf0,v(tg)𝑑t.subscript𝜑0𝑣𝑔subscript\subscript𝑍𝑣subscript𝔗Δ𝑣subscript𝑓0𝑣𝑡𝑔differential-d𝑡\varphi_{0,v}(g)=\textstyle{\int}_{Z_{v}\backslash{\mathfrak{T}}_{\Delta,v}}f_{0,v}(tg)dt.

Then we have

φ0,v(12)=|mv|v1×{1(vΣ),qvdv/2(vΣfinΣdyadicandΔv0𝔬v×(𝔬v×)2),qvdv/2312(1+2z)(vΣdyadic,Δv0=5),qvdv/2(1+qv(z+1)/2)(vΣfin,Δv0𝔭v𝔭v2),qvdv/2(1+2(z+1)/2)(vΣdyadic,Δv0{5,1}).subscript𝜑0𝑣subscript12superscriptsubscriptsubscript𝑚𝑣𝑣1cases1𝑣subscriptΣsuperscriptsubscript𝑞𝑣subscript𝑑𝑣2𝑣subscriptΣfinsubscriptΣdyadicandsuperscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2superscriptsubscript𝑞𝑣subscript𝑑𝑣2superscript3121superscript2𝑧formulae-sequence𝑣subscriptΣdyadicsuperscriptsubscriptΔ𝑣05superscriptsubscript𝑞𝑣subscript𝑑𝑣21superscriptsubscript𝑞𝑣𝑧12formulae-sequence𝑣subscriptΣfinsuperscriptsubscriptΔ𝑣0subscript𝔭𝑣superscriptsubscript𝔭𝑣2superscriptsubscript𝑞𝑣subscript𝑑𝑣21superscript2𝑧12formulae-sequence𝑣subscriptΣdyadicsuperscriptsubscriptΔ𝑣051\varphi_{0,v}(1_{2})=|m_{v}|_{v}^{-1}\times\begin{cases}1&(v\in\Sigma_{\infty}),\\ q_{v}^{-d_{v}/2}&(v\in\Sigma_{\rm fin}-\Sigma_{\rm dyadic}\,\text{and}\,\Delta_{v}^{0}\in{\mathfrak{o}}_{v}^{\times}-({\mathfrak{o}}_{v}^{\times})^{2}),\\ q_{v}^{-d_{v}/2}3^{-1}2(1+2^{-z})&(v\in\Sigma_{\rm dyadic},\,\Delta_{v}^{0}=5),\\ q_{v}^{-d_{v}/2}(1+q_{v}^{(z+1)/2})&(v\in\Sigma_{\rm fin},\,\Delta_{v}^{0}\in{\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2}),\\ q_{v}^{-d_{v}/2}(1+2^{-(z+1)/2})&(v\in\Sigma_{\rm dyadic},\,\Delta_{v}^{0}\in\{-5,-1\}).\end{cases}
Proof.

We have the values of φ0,v(12)subscript𝜑0𝑣subscript12\varphi_{0,v}(1_{2}) by Lemmas 7.2 and 7.3 easily. ∎

Lemma 7.5.

Let vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}. Then there exists a unique smooth function φvΔ,(z):Gv:superscriptsubscript𝜑𝑣Δ𝑧subscript𝐺𝑣\varphi_{v}^{\Delta,(z)}:G_{v}\rightarrow{\mathbb{C}} such that φvΔ,(z)(12)=1superscriptsubscript𝜑𝑣Δ𝑧subscript121\varphi_{v}^{\Delta,(z)}(1_{2})=1 with the properties:

  • (i)

    φvΔ,(z)(hgk)=φvΔ,(z)(g)superscriptsubscript𝜑𝑣Δ𝑧𝑔𝑘superscriptsubscript𝜑𝑣Δ𝑧𝑔\varphi_{v}^{\Delta,(z)}(hgk)=\varphi_{v}^{\Delta,(z)}(g) for all h𝔗Δ,vsubscript𝔗Δ𝑣h\in{\mathfrak{T}}_{\Delta,v} and k𝕂v𝑘subscript𝕂𝑣k\in{\mathbb{K}}_{v}.

  • (ii)

    R(𝕋v)φvΔ,(z)=(qv1+z2+qv1z2)φvΔ,(z)𝑅subscript𝕋𝑣superscriptsubscript𝜑𝑣Δ𝑧superscriptsubscript𝑞𝑣1𝑧2superscriptsubscript𝑞𝑣1𝑧2superscriptsubscript𝜑𝑣Δ𝑧R({\mathbb{T}}_{v})\varphi_{v}^{\Delta,(z)}=(q_{v}^{\frac{1+z}{2}}+q_{v}^{\frac{1-z}{2}})\,\varphi_{v}^{\Delta,(z)} if vΣfin𝑣subscriptΣfinv\in\Sigma_{{\rm fin}}.

  • (ii)’

    R(Ωv)φvΔ,(z)=z212φvΔ,(z)𝑅subscriptΩ𝑣superscriptsubscript𝜑𝑣Δ𝑧superscript𝑧212superscriptsubscript𝜑𝑣Δ𝑧R(\Omega_{v})\varphi_{v}^{\Delta,(z)}=\frac{z^{2}-1}{2}\varphi_{v}^{\Delta,(z)} if vΣ𝑣subscriptΣv\in\Sigma_{\infty}.

Proof.

If Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1, then 𝔗Δ,v=Hvsubscript𝔗Δ𝑣subscript𝐻𝑣{\mathfrak{T}}_{\Delta,v}=H_{v}. Thus, the assertion follows from Lemma 6.1. If Δv01superscriptsubscriptΔ𝑣01\Delta_{v}^{0}\not=1, then the uniqueness of φvΔ,(z)superscriptsubscript𝜑𝑣Δ𝑧\varphi_{v}^{\Delta,(z)} follows from [34] ([32, Proposition 9’]). From Lemma 7.4, φ0,v(12)0subscript𝜑0𝑣subscript120\varphi_{0,v}(1_{2})\not=0. Then the function φvΔ,(z)(g)=φ0,v(12)1φ0,v(g)superscriptsubscript𝜑𝑣Δ𝑧𝑔subscript𝜑0𝑣superscriptsubscript121subscript𝜑0𝑣𝑔\varphi_{v}^{\Delta,(z)}(g)=\varphi_{0,v}(1_{2})^{-1}\varphi_{0,v}(g) meets all the requirements. ∎

By exchanging the order of integrals,

(7.5) (β)Δ(g)=Lσβ(z)ΛF(z+1)EΔ(z;g)𝑑zsuperscriptsuperscriptsubscript𝛽Δ𝑔subscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧1superscript𝐸Δ𝑧𝑔differential-d𝑧\displaystyle({\mathcal{E}}_{\beta}^{*})^{\Delta}(g)=\int_{L_{\sigma}}\beta(z)\Lambda_{F}(z+1)\,E^{\Delta}(z;g)\,{{d}}z

with

EΔ(z;g)=𝔸×E×\𝔸E×E(z;ιΔ(τ)RΔ1g)d×τ,z.formulae-sequencesuperscript𝐸Δ𝑧𝑔subscript\superscript𝔸superscript𝐸superscriptsubscript𝔸𝐸𝐸𝑧subscript𝜄Δ𝜏superscriptsubscript𝑅Δ1𝑔superscript𝑑𝜏𝑧\displaystyle E^{\Delta}(z;g)=\int_{{\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times}}E\left(z;\iota_{\Delta}(\tau)R_{\Delta}^{-1}g\right)\,{{d}}^{\times}\tau,\quad z\in{\mathbb{C}}.

From Lemma 7.5, this is decomposed into a product as

(7.6) EΔ(z;g)=EΔ(z;12)vΣFφvΔ,(z)(gv),gG𝔸.formulae-sequencesuperscript𝐸Δ𝑧𝑔superscript𝐸Δ𝑧subscript12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣Δ𝑧subscript𝑔𝑣𝑔subscript𝐺𝔸\displaystyle E^{\Delta}(z;g)=E^{\Delta}(z;1_{2})\,\prod_{v\in\Sigma_{F}}\varphi_{v}^{\Delta,(z)}(g_{v}),\quad g\in G_{\mathbb{A}}.

Let Re(z)>1Re𝑧1\operatorname{Re}(z)>1 and set T=ιΔ(E×)𝑇subscript𝜄Δsuperscript𝐸T=\iota_{\Delta}(E^{\times}). Since BT=G𝐵𝑇𝐺BT=G, by noting TB=Z𝑇𝐵𝑍T\cap B=Z, we have

(7.7) EΔ(z;12)=superscript𝐸Δ𝑧subscript12absent\displaystyle E^{\Delta}(z;1_{2})= Z𝔸\T𝔸y(hRΔ1)(z+1)/2𝑑h=vΣFYv(z)withYv(z)=Zv\Tvy(hvRΔ,v1)(z+1)/2𝑑hv.formulae-sequencesubscript\subscript𝑍𝔸subscript𝑇𝔸𝑦superscriptsuperscriptsubscript𝑅Δ1𝑧12differential-dsubscriptproduct𝑣subscriptΣ𝐹subscript𝑌𝑣𝑧withsubscript𝑌𝑣𝑧subscript\subscript𝑍𝑣subscript𝑇𝑣𝑦superscriptsubscript𝑣superscriptsubscript𝑅Δ𝑣1𝑧12differential-dsubscript𝑣\displaystyle\int_{Z_{\mathbb{A}}\backslash T_{\mathbb{A}}}y(hR_{\Delta}^{-1})^{(z+1)/2}dh=\prod_{v\in\Sigma_{F}}Y_{v}(z)\quad\text{with}\quad Y_{v}(z)=\int_{Z_{v}\backslash T_{v}}y(h_{v}R_{\Delta,v}^{-1})^{(z+1)/2}\,dh_{v}.
Lemma 7.6.

Let D={vΣdyadic|Δv0=5}𝐷conditional-set𝑣subscriptΣdyadicsuperscriptsubscriptΔ𝑣05D=\{v\in\Sigma_{\rm dyadic}|\,\Delta_{v}^{0}=5\,\} and ΣsplitΔ={vΣfin|Δv0=1}superscriptsubscriptΣsplitΔconditional-set𝑣subscriptΣfinsuperscriptsubscriptΔ𝑣01\Sigma_{\rm split}^{\Delta}=\{v\in\Sigma_{\rm fin}|\,\Delta_{v}^{0}=1\}. Then

Yv(z)=|mv|v1d(v)|Δv0mv|v(z+1)/2ζEv(z+12)ζFv(z+1)1×{|2|v(z1)/2(vΣsplitΔΣ),1(vΣfinDΣsplitΔ),23(1+2z)(vD),subscript𝑌𝑣𝑧superscriptsubscriptsubscript𝑚𝑣𝑣1𝑑𝑣superscriptsubscriptsuperscriptsubscriptΔ𝑣0subscript𝑚𝑣𝑣𝑧12subscript𝜁subscript𝐸𝑣𝑧12subscript𝜁subscript𝐹𝑣superscript𝑧11casessuperscriptsubscript2𝑣𝑧12𝑣superscriptsubscriptΣsplitΔsubscriptΣotherwise1𝑣subscriptΣfin𝐷superscriptsubscriptΣsplitΔotherwise231superscript2𝑧𝑣𝐷otherwise\displaystyle Y_{v}(z)=|m_{v}|_{v}^{-1}d(v)|\Delta_{v}^{0}m_{v}|_{v}^{-(z+1)/2}\zeta_{E_{v}}\left(\tfrac{z+1}{2}\right){\zeta_{F_{v}}(z+1)^{-1}}\times\begin{cases}|2|_{v}^{(z-1)/2}\quad(v\in\Sigma_{\rm split}^{\Delta}\cup\Sigma_{\infty}),\\ 1\quad(v\in\Sigma_{\rm fin}-D\cup\Sigma_{\rm split}^{\Delta}),\\ \tfrac{2}{3}(1+2^{-z})\quad(v\in D),\end{cases}

where we set d(v)=qvdv/2𝑑𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣2d(v)=q_{v}^{-d_{v}/2} for vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and d(v)=1𝑑𝑣1d(v)=1 for vΣ𝑣subscriptΣv\in\Sigma_{\infty}.

Proof.

Let us recall the formula

y(gv)(z+1)/2=d(v)1ζF,v(z+1)1|det(gv)|v(z+1)/2Fv×|a|vz+1Φv(a[0,1]g)d×a,𝑦superscriptsubscript𝑔𝑣𝑧12𝑑superscript𝑣1subscript𝜁𝐹𝑣superscript𝑧11superscriptsubscriptsubscript𝑔𝑣𝑣𝑧12subscriptsuperscriptsubscript𝐹𝑣superscriptsubscript𝑎𝑣𝑧1subscriptΦ𝑣𝑎01𝑔superscript𝑑𝑎y(g_{v})^{(z+1)/2}={d(v)^{-1}}{\zeta_{F,v}(z+1)^{-1}}|\det(g_{v})|_{v}^{(z+1)/2}\textstyle{\int}_{F_{v}^{\times}}|a|_{v}^{z+1}\Phi_{v}(a[0,1]g)d^{\times}a,

where Φv(x,y)=eπ(x2+y2)subscriptΦ𝑣𝑥𝑦superscript𝑒𝜋superscript𝑥2superscript𝑦2\Phi_{v}(x,y)=e^{-\pi(x^{2}+y^{2})} if vΣ𝑣subscriptΣv\in\Sigma_{\infty} and Φv(x,y)=ch𝔬v×𝔬v(x,y)subscriptΦ𝑣𝑥𝑦subscriptchsubscript𝔬𝑣subscript𝔬𝑣𝑥𝑦\Phi_{v}(x,y)={\rm ch}_{{\mathfrak{o}}_{v}\times{\mathfrak{o}}_{v}}(x,y) if vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. Noting det(ιΔ(τ))=NE/F(τ)subscript𝜄Δ𝜏subscriptN𝐸𝐹𝜏\det(\iota_{\Delta}(\tau))={\operatorname{N}}_{E/F}(\tau), we have

Yv(z)=d(v)1|det(RΔ,v1)|v(z+1)/2ζFv(z+1)1Yv0(z)subscript𝑌𝑣𝑧𝑑superscript𝑣1superscriptsubscriptsuperscriptsubscript𝑅Δ𝑣1𝑣𝑧12subscript𝜁subscript𝐹𝑣superscript𝑧11superscriptsubscript𝑌𝑣0𝑧\displaystyle Y_{v}(z)={d(v)^{-1}|\det(R_{\Delta,v}^{-1})|_{v}^{(z+1)/2}}{\zeta_{F_{v}}(z+1)^{-1}}\,Y_{v}^{0}(z)

with

Yv0(z)=Fv×\Ev×|NEv/Fv(τ)|vz+12{Fv×|a|vz+1Φv(a[0,1][xy41Δyx]RΔ,v1)d×a}d×τ,superscriptsubscript𝑌𝑣0𝑧subscript\superscriptsubscript𝐹𝑣superscriptsubscript𝐸𝑣superscriptsubscriptsubscriptNsubscript𝐸𝑣subscript𝐹𝑣𝜏𝑣𝑧12conditional-setsubscriptsuperscriptsubscript𝐹𝑣evaluated-at𝑎𝑣𝑧1subscriptΦ𝑣𝑎01delimited-[]𝑥𝑦superscript41Δ𝑦𝑥superscriptsubscript𝑅Δ𝑣1superscript𝑑𝑎superscript𝑑𝜏Y_{v}^{0}(z)=\textstyle{\int}_{F_{v}^{\times}\backslash E_{v}^{\times}}|{\operatorname{N}}_{E_{v}/F_{v}}(\tau)|_{v}^{\frac{z+1}{2}}\{\int_{F_{v}^{\times}}|a|_{v}^{z+1}\Phi_{v}\biggl{(}a[0,1]\left[\begin{smallmatrix}x&y\\ 4^{-1}\Delta y&x\end{smallmatrix}\right]R_{\Delta,v}^{-1}\biggr{)}{{d}}^{\times}a\}\,{{d}}^{\times}\tau,

where τ=x+yΔ/2𝜏𝑥𝑦Δ2\tau=x+y\sqrt{\Delta}/2 and d×τ=ζEv(1)|x241Δy2|v1dxdysuperscript𝑑𝜏subscript𝜁subscript𝐸𝑣1superscriptsubscriptsuperscript𝑥2superscript41Δsuperscript𝑦2𝑣1𝑑𝑥𝑑𝑦{{d}}^{\times}\tau=\zeta_{E_{v}}(1)|x^{2}-4^{-1}\Delta y^{2}|_{v}^{-1}\,{{d}}x\,{{d}}y.

(i) Suppose vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1. Then we have

Yv0(z)=superscriptsubscript𝑌𝑣0𝑧absent\displaystyle Y_{v}^{0}(z)= |mv|v1ζEv(1)Fv××Fv×|4XY|v(z+1)/2Φv([X,Y])|2XY|v1𝑑X𝑑Y=|mv|v1|4|vz2qvdvζEv(z+12),superscriptsubscriptsubscript𝑚𝑣𝑣1subscript𝜁subscript𝐸𝑣1subscriptsuperscriptsubscript𝐹𝑣superscriptsubscript𝐹𝑣superscriptsubscript4𝑋𝑌𝑣𝑧12subscriptΦ𝑣𝑋𝑌superscriptsubscript2𝑋𝑌𝑣1differential-d𝑋differential-d𝑌superscriptsubscriptsubscript𝑚𝑣𝑣1superscriptsubscript4𝑣𝑧2superscriptsubscript𝑞𝑣subscript𝑑𝑣subscript𝜁subscript𝐸𝑣𝑧12\displaystyle|m_{v}|_{v}^{-1}\zeta_{E_{v}}(1)\textstyle{\int}_{F_{v}^{\times}\times F_{v}^{\times}}|4XY|_{v}^{(z+1)/2}\Phi_{v}([X,Y])\,|2XY|_{v}^{-1}\,{{d}}X\,{{d}}Y=|m_{v}|_{v}^{-1}\,|4|_{v}^{\frac{z}{2}}q_{v}^{-d_{v}}\zeta_{E_{v}}\left(\tfrac{z+1}{2}\right),

by the variable change X=a(mvy+x)2𝑋𝑎subscript𝑚𝑣𝑦𝑥2X=\frac{a(m_{v}y+x)}{2}, Y=a(mvyx)2𝑌𝑎subscript𝑚𝑣𝑦𝑥2Y=\frac{a(m_{v}y-x)}{2} and by the relation Φv([X,Y])=δ(X𝔬v)δ(Y𝔬v)subscriptΦ𝑣𝑋𝑌𝛿𝑋subscript𝔬𝑣𝛿𝑌subscript𝔬𝑣\Phi_{v}([X,Y])=\delta(X\in\mathfrak{o}_{v})\delta(Y\in\mathfrak{o}_{v}).

(ii) Suppose vΣfinTD𝑣subscriptΣfin𝑇𝐷v\in\Sigma_{\rm fin}-T\cup D. Then we easily have

Yv0(z)=superscriptsubscript𝑌𝑣0𝑧absent\displaystyle Y_{v}^{0}(z)= |mv|v1Ev×|NEv/Fv(τ0)|v(z+1)/2Φv([Δv0y0,x])d×τ0,superscriptsubscriptsubscript𝑚𝑣𝑣1subscriptsuperscriptsubscript𝐸𝑣superscriptsubscriptsubscriptNsubscript𝐸𝑣subscript𝐹𝑣subscript𝜏0𝑣𝑧12subscriptΦ𝑣superscriptsubscriptΔ𝑣0subscript𝑦0𝑥superscript𝑑subscript𝜏0\displaystyle|m_{v}|_{v}^{-1}\textstyle{\int}_{E_{v}^{\times}}|{\operatorname{N}}_{E_{v}/F_{v}}(\tau_{0})|_{v}^{(z+1)/2}\Phi_{v}([\Delta_{v}^{0}y_{0},x])\,{{d}}^{\times}\tau_{0},

with y0=mvysubscript𝑦0subscript𝑚𝑣𝑦y_{0}=m_{v}y and τ0=x+Δv0y0subscript𝜏0𝑥superscriptsubscriptΔ𝑣0subscript𝑦0\tau_{0}=x+\sqrt{\Delta_{v}^{0}}\,y_{0}. From Φv([Δv0y0,x])=δ(x𝔬v)δ(y0(Δv0)1𝔬v)subscriptΦ𝑣superscriptsubscriptΔ𝑣0subscript𝑦0𝑥𝛿𝑥subscript𝔬𝑣𝛿subscript𝑦0superscriptsuperscriptsubscriptΔ𝑣01subscript𝔬𝑣\Phi_{v}([\Delta_{v}^{0}\,y_{0},x])=\delta(x\in\mathfrak{o}_{v})\delta(y_{0}\in(\Delta_{v}^{0})^{-1}\mathfrak{o}_{v}),

Yv0(z)=superscriptsubscript𝑌𝑣0𝑧absent\displaystyle Y_{v}^{0}(z)= |mv|v1x𝔬v{0}y0𝔬v/Δv0{0}|x2Δv0y02|vz+121ζEv(1)𝑑x𝑑y0superscriptsubscriptsubscript𝑚𝑣𝑣1subscript𝑥subscript𝔬𝑣0subscriptsubscript𝑦0subscript𝔬𝑣superscriptsubscriptΔ𝑣00superscriptsubscriptsuperscript𝑥2superscriptsubscriptΔ𝑣0superscriptsubscript𝑦02𝑣𝑧121subscript𝜁subscript𝐸𝑣1differential-d𝑥differential-dsubscript𝑦0\displaystyle|m_{v}|_{v}^{-1}\textstyle{\int}_{x\in\mathfrak{o}_{v}-\{0\}}\int_{y_{0}\in\mathfrak{o}_{v}/\Delta_{v}^{0}-\{0\}}|x^{2}-\Delta_{v}^{0}\,y_{0}^{2}|_{v}^{\frac{z+1}{2}-1}\zeta_{E_{v}}(1)\,{{d}}x\,{{d}}y_{0}
=\displaystyle= |mv|v1|Δv0|v(z+1)/2𝔬v2{(0,0)}|y2Δv0x2|vz12ζEv(1)𝑑x𝑑ysuperscriptsubscriptsubscript𝑚𝑣𝑣1superscriptsubscriptsuperscriptsubscriptΔ𝑣0𝑣𝑧12subscriptdouble-integralsuperscriptsubscript𝔬𝑣200superscriptsubscriptsuperscript𝑦2superscriptsubscriptΔ𝑣0superscript𝑥2𝑣𝑧12subscript𝜁subscript𝐸𝑣1differential-d𝑥differential-d𝑦\displaystyle|m_{v}|_{v}^{-1}|\Delta_{v}^{0}|_{v}^{-(z+1)/2}{\textstyle\iint}_{\mathfrak{o}_{v}^{2}-\{(0,0)\}}|y^{2}-\Delta_{v}^{0}\,x^{2}|_{v}^{\frac{z-1}{2}}\zeta_{E_{v}}(1){{d}}x\,{{d}}y
=\displaystyle= |mv|v1|Δv0|vz+12vol(𝔬Ev×,d×τ)ζEv(z+12)superscriptsubscriptsubscript𝑚𝑣𝑣1superscriptsubscriptsuperscriptsubscriptΔ𝑣0𝑣𝑧12volsuperscriptsubscript𝔬subscript𝐸𝑣superscript𝑑𝜏subscript𝜁subscript𝐸𝑣𝑧12\displaystyle|m_{v}|_{v}^{-1}|{\Delta_{v}^{0}}|_{v}^{-\frac{z+1}{2}}\,{\operatorname{vol}}({\mathfrak{o}}_{E_{v}}^{\times},d^{\times}\tau)\zeta_{E_{v}}\left(\tfrac{z+1}{2}\right)

by making the variable change y=Δv0y0𝑦superscriptsubscriptΔ𝑣0subscript𝑦0y=\Delta_{v}^{0}y_{0} and noting 𝔬Ev=𝔬v+Δv0𝔬vsubscript𝔬subscript𝐸𝑣subscript𝔬𝑣superscriptsubscriptΔ𝑣0subscript𝔬𝑣{\mathfrak{o}}_{E_{v}}={\mathfrak{o}}_{v}+\sqrt{\Delta_{v}^{0}}{\mathfrak{o}}_{v}.

(iii) Let vD𝑣𝐷v\in D. By the same computation as in (ii), we have

Yv0(z)=|mv|1𝔬v{0}𝔬v{0}|y2Δv0x2|v(z1)/2ζEv(1)𝑑x𝑑y.superscriptsubscript𝑌𝑣0𝑧superscriptsubscript𝑚𝑣1subscriptsubscript𝔬𝑣0subscriptsubscript𝔬𝑣0superscriptsubscriptsuperscript𝑦2superscriptsubscriptΔ𝑣0superscript𝑥2𝑣𝑧12subscript𝜁subscript𝐸𝑣1differential-d𝑥differential-d𝑦Y_{v}^{0}(z)=|m_{v}|^{-1}\textstyle{\int}_{{\mathfrak{o}}_{v}-\{0\}}\textstyle{\int}_{{\mathfrak{o}}_{v}-\{0\}}|y^{2}-\Delta_{v}^{0}x^{2}|_{v}^{(z-1)/2}\zeta_{E_{v}}(1)dxdy.

We decompose the set (𝔬v{0})×(𝔬v{0})subscript𝔬𝑣0subscript𝔬𝑣0(\mathfrak{o}_{v}-\{0\})\times(\mathfrak{o}_{v}-\{0\}) into the disjoint union of D1=(𝔭v{0})×(𝔬v{0})subscript𝐷1subscript𝔭𝑣0subscript𝔬𝑣0D_{1}=({\mathfrak{p}}_{v}-\{0\})\times(\mathfrak{o}_{v}-\{0\}), D2=𝔬v××(𝔭v{0})subscript𝐷2superscriptsubscript𝔬𝑣subscript𝔭𝑣0D_{2}=\mathfrak{o}_{v}^{\times}\times({\mathfrak{p}}_{v}-\{0\}) and D3=𝔬v××𝔬v×subscript𝐷3superscriptsubscript𝔬𝑣superscriptsubscript𝔬𝑣D_{3}=\mathfrak{o}_{v}^{\times}\times\mathfrak{o}_{v}^{\times} and write |mv|vYv(z)=I1+I2+I3subscriptsubscript𝑚𝑣𝑣subscript𝑌𝑣𝑧subscript𝐼1subscript𝐼2subscript𝐼3|m_{v}|_{v}\,Y_{v}(z)=I_{1}+I_{2}+I_{3}, where Ii=Di|y2Δv0x2|v(z1)/2ζEv(1)𝑑x𝑑ysubscript𝐼𝑖subscriptdouble-integralsubscript𝐷𝑖superscriptsubscriptsuperscript𝑦2superscriptsubscriptΔ𝑣0superscript𝑥2𝑣𝑧12subscript𝜁subscript𝐸𝑣1differential-d𝑥differential-d𝑦I_{i}=\iint_{D_{i}}|y^{2}-\Delta_{v}^{0}x^{2}|_{v}^{(z-1)/2}\zeta_{E_{v}}(1)dxdy with i=1,2,3𝑖123i=1,2,3. We have

I1=subscript𝐼1absent\displaystyle I_{1}= (𝔭v{0}𝔬v×+𝔭v{0}𝔭v{0})|y2Δv0x2|v(z1)/2ζEv(1)dxdysubscriptsubscript𝔭𝑣0subscriptsuperscriptsubscript𝔬𝑣subscriptsubscript𝔭𝑣0subscriptsubscript𝔭𝑣0superscriptsubscriptsuperscript𝑦2superscriptsubscriptΔ𝑣0superscript𝑥2𝑣𝑧12subscript𝜁subscript𝐸𝑣1𝑑𝑥𝑑𝑦\displaystyle\left(\textstyle{\int}_{{\mathfrak{p}}_{v}-\{0\}}\textstyle{\int}_{{\mathfrak{o}}_{v}^{\times}}+\textstyle{\int}_{{\mathfrak{p}}_{v}-\{0\}}\int_{{\mathfrak{p}}_{v}-\{0\}}\right)|y^{2}-\Delta_{v}^{0}x^{2}|_{v}^{(z-1)/2}\zeta_{E_{v}}(1)dxdy
=\displaystyle= (1+qv)1qvdv+qvz1|mv|vYv0(z),superscript1subscript𝑞𝑣1superscriptsubscript𝑞𝑣subscript𝑑𝑣superscriptsubscript𝑞𝑣𝑧1subscriptsubscript𝑚𝑣𝑣superscriptsubscript𝑌𝑣0𝑧\displaystyle(1+q_{v})^{-1}q_{v}^{-d_{v}}+q_{v}^{-z-1}|m_{v}|_{v}\,Y_{v}^{0}(z),

and I2=(1+qv)1qvdvsubscript𝐼2superscript1subscript𝑞𝑣1superscriptsubscript𝑞𝑣subscript𝑑𝑣I_{2}=(1+q_{v})^{-1}q_{v}^{-d_{v}} easily. Since vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}, we have that u𝔬v×𝑢superscriptsubscript𝔬𝑣u\in\mathfrak{o}_{v}^{\times} is a square if and only if u(mod4𝔭v)annotated𝑢pmod4subscript𝔭𝑣u\pmod{4{\mathfrak{p}}_{v}} is a square in 𝔬v×/1+4𝔭vsuperscriptsubscript𝔬𝑣14subscript𝔭𝑣\mathfrak{o}_{v}^{\times}/1+4{\mathfrak{p}}_{v}. In particular, ξ2Δv04𝔭vsuperscript𝜉2superscriptsubscriptΔ𝑣04subscript𝔭𝑣\xi^{2}-\Delta_{v}^{0}\notin 4{\mathfrak{p}}_{v} for ξ𝔬v×𝜉superscriptsubscript𝔬𝑣\xi\in\mathfrak{o}_{v}^{\times}. By this remark, we compute

I3subscript𝐼3\displaystyle I_{3} =𝔬v×𝔬v×|x2y2Δv0x2|v(z1)/2ζEv(1)𝑑x𝑑y=(1+qv1)1qvdv/2𝔬v×|y2Δv0|v(z1)/2𝑑yabsentsubscriptsuperscriptsubscript𝔬𝑣subscriptsuperscriptsubscript𝔬𝑣superscriptsubscriptsuperscript𝑥2superscript𝑦2superscriptsubscriptΔ𝑣0superscript𝑥2𝑣𝑧12subscript𝜁subscript𝐸𝑣1differential-d𝑥differential-d𝑦superscript1superscriptsubscript𝑞𝑣11superscriptsubscript𝑞𝑣subscript𝑑𝑣2subscriptsuperscriptsubscript𝔬𝑣superscriptsubscriptsuperscript𝑦2superscriptsubscriptΔ𝑣0𝑣𝑧12differential-d𝑦\displaystyle=\textstyle{\int}_{{\mathfrak{o}}_{v}^{\times}}\textstyle{\int}_{{\mathfrak{o}}_{v}^{\times}}|x^{2}y^{2}-\Delta_{v}^{0}x^{2}|_{v}^{(z-1)/2}\zeta_{E_{v}}(1)dxdy=(1+q_{v}^{-1})^{-1}q_{v}^{-d_{v}/2}\textstyle{\int}_{{\mathfrak{o}}_{v}^{\times}}|y^{2}-\Delta_{v}^{0}|_{v}^{(z-1)/2}dy
=(1+qv1)1qvdv/2ξ𝔬v×/1+4𝔭v𝔬v|{ξ(1+4ϖvu)}2Δv0|v(z1)/2|4ϖv|v𝑑uabsentsuperscript1superscriptsubscript𝑞𝑣11superscriptsubscript𝑞𝑣subscript𝑑𝑣2subscript𝜉superscriptsubscript𝔬𝑣14subscript𝔭𝑣subscriptsubscript𝔬𝑣superscriptsubscriptsuperscript𝜉14subscriptitalic-ϖ𝑣𝑢2superscriptsubscriptΔ𝑣0𝑣𝑧12subscript4subscriptitalic-ϖ𝑣𝑣differential-d𝑢\displaystyle=(1+q_{v}^{-1})^{-1}q_{v}^{-d_{v}/2}\sum_{\xi\in{\mathfrak{o}}_{v}^{\times}/1+4{\mathfrak{p}}_{v}}\textstyle{\int}_{{\mathfrak{o}}_{v}}|\{\xi(1+4\varpi_{v}u)\}^{2}-\Delta_{v}^{0}|_{v}^{(z-1)/2}|4\varpi_{v}|_{v}du
=(1+qv1)1qvdv|4ϖv|vξ𝔬v×/1+4𝔭v|ξ2Δv0|v(z1)/2=(1+qv1)1qvdv|4|vqv1j=0ordv(4)u(j)qvj(z1)/2,absentsuperscript1superscriptsubscript𝑞𝑣11superscriptsubscript𝑞𝑣subscript𝑑𝑣subscript4subscriptitalic-ϖ𝑣𝑣subscript𝜉superscriptsubscript𝔬𝑣14subscript𝔭𝑣superscriptsubscriptsuperscript𝜉2superscriptsubscriptΔ𝑣0𝑣𝑧12superscript1superscriptsubscript𝑞𝑣11superscriptsubscript𝑞𝑣subscript𝑑𝑣subscript4𝑣superscriptsubscript𝑞𝑣1superscriptsubscript𝑗0subscriptord𝑣4𝑢𝑗superscriptsubscript𝑞𝑣𝑗𝑧12\displaystyle=(1+q_{v}^{-1})^{-1}q_{v}^{-d_{v}}|4\varpi_{v}|_{v}\sum_{\xi\in{\mathfrak{o}}_{v}^{\times}/1+4{\mathfrak{p}}_{v}}|\xi^{2}-\Delta_{v}^{0}|_{v}^{(z-1)/2}=(1+q_{v}^{-1})^{-1}q_{v}^{-d_{v}}|4|_{v}q_{v}^{-1}\sum_{j=0}^{\operatorname{ord}_{v}(4)}u(j)q_{v}^{-j(z-1)/2},

where u(j)𝑢𝑗u(j) is the number of all ξ𝔬v×/(1+4𝔭v)𝜉superscriptsubscript𝔬𝑣14subscript𝔭𝑣\xi\in{\mathfrak{o}}_{v}^{\times}/(1+4{\mathfrak{p}}_{v}) such that ordv(ξ2Δv0)=jsubscriptord𝑣superscript𝜉2superscriptsubscriptΔ𝑣0𝑗\operatorname{ord}_{v}(\xi^{2}-\Delta_{v}^{0})=j. Hence

|mv|vYv0(z)=ζEv(z+12){2(1+qv)1+(1+qv1)1|4|vqv1j=0ordv(4)u(j)qvj(z1)/2}=ζEv(z+12)23(1+2z)subscriptsubscript𝑚𝑣𝑣superscriptsubscript𝑌𝑣0𝑧subscript𝜁subscript𝐸𝑣𝑧12conditional-set2superscript1subscript𝑞𝑣1superscript1superscriptsubscript𝑞𝑣11evaluated-at4𝑣superscriptsubscript𝑞𝑣1superscriptsubscript𝑗0subscriptord𝑣4𝑢𝑗superscriptsubscript𝑞𝑣𝑗𝑧12subscript𝜁subscript𝐸𝑣𝑧12231superscript2𝑧|m_{v}|_{v}Y_{v}^{0}(z)=\zeta_{E_{v}}(\tfrac{z+1}{2})\{2(1+q_{v})^{-1}+(1+q_{v}^{-1})^{-1}|4|_{v}q_{v}^{-1}\sum_{j=0}^{\operatorname{ord}_{v}(4)}u(j)q_{v}^{-j(z-1)/2}\}=\zeta_{E_{v}}(\tfrac{z+1}{2})\tfrac{2}{3}(1+2^{-z})

by using qv=2subscript𝑞𝑣2q_{v}=2, ordv(4)=2subscriptord𝑣42\operatorname{ord}_{v}(4)=2, u(0)=u(1)=0𝑢0𝑢10u(0)=u(1)=0 and u(2)=4𝑢24u(2)=4. We omit the detail of the computation for archimedean cases (iii) and (iv), which are elementary.∎

The following formula is originally due to Hecke ([20, Chap. II, §3]).

Proposition 7.7.

We have

EΔ(z;12)=superscript𝐸Δ𝑧subscript12absent\displaystyle E^{\Delta}(z;1_{2})= {vΣF|mv|v1}{vΣFΔv0=1|2|v1}DF1/2N(𝔡E/F)z+14ζF((z+1)/2)L((z+1)/2,εΔ)ζF(z+1)\displaystyle\{\prod_{v\in\Sigma_{F}}|m_{v}|_{v}^{-1}\}\,\{\prod_{\begin{subarray}{c}v\in\Sigma_{F}\\ \Delta_{v}^{0}=1\end{subarray}}|2|_{v}^{-1}\}\,D_{F}^{-1/2}\,{\operatorname{N}}({\mathfrak{d}}_{E/F})^{\frac{z+1}{4}}\,\frac{\zeta_{F}((z+1)/2)L((z+1)/2,\varepsilon_{\Delta})}{\zeta_{F}(z+1)}
×vΣΔv0=121vΣdyadicΔv0=5312z+12+1(1+2z),\displaystyle\times\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ \Delta^{0}_{v}=-1\end{subarray}}2^{-1}\,\prod_{\begin{subarray}{c}v\in\Sigma_{\rm{dyadic}}\\ \Delta_{v}^{0}=5\end{subarray}}{3^{-1}}2^{\frac{z+1}{2}+1}(1+2^{-z}),

where E=F(Δ)𝐸𝐹ΔE=F(\sqrt{\Delta}) and 𝔡E/Fsubscript𝔡𝐸𝐹{\mathfrak{d}}_{E/F} denotes the relative discriminant of E/F𝐸𝐹E/F.

Proof.

From Lemma 7.6 and (7.7), we have that EΔ(z;12)superscript𝐸Δ𝑧subscript12E^{\Delta}(z;1_{2}) with Re(z)>1Re𝑧1\operatorname{Re}(z)>1 is the product of ζE(z+12)ζF(1+z)1subscript𝜁𝐸𝑧12subscript𝜁𝐹superscript1𝑧1\zeta_{E}(\frac{z+1}{2})\zeta_{F}(1+z)^{-1} and finite factors

(7.8) vΣF|mv|v1×DF1/2×vΣF|mvΔv0|vz+12×vΣsplitΔΣ|2|vz12×vD23(1+2z).subscriptproduct𝑣subscriptΣ𝐹superscriptsubscriptsubscript𝑚𝑣𝑣1superscriptsubscript𝐷𝐹12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscriptsubscript𝑚𝑣superscriptsubscriptΔ𝑣0𝑣𝑧12subscriptproduct𝑣superscriptsubscriptΣsplitΔsubscriptΣsuperscriptsubscript2𝑣𝑧12subscriptproduct𝑣𝐷231superscript2𝑧\displaystyle\prod_{v\in\Sigma_{F}}|m_{v}|_{v}^{-1}\times D_{F}^{-1/2}\times\prod_{v\in\Sigma_{F}}|m_{v}\Delta_{v}^{0}|_{v}^{-\frac{z+1}{2}}\times\prod_{v\in\Sigma_{\rm split}^{\Delta}\cup\Sigma_{\infty}}|2|_{v}^{\frac{z-1}{2}}\times\prod_{v\in D}\tfrac{2}{3}(1+2^{-z}).

By 1=vΣF|41Δ|v=vΣF|Δv0|v|mv|v2=vΣF|Δv0|vvΣF|mv|v21subscriptproduct𝑣subscriptΣ𝐹subscriptsuperscript41Δ𝑣subscriptproduct𝑣subscriptΣ𝐹subscriptsuperscriptsubscriptΔ𝑣0𝑣superscriptsubscriptsubscript𝑚𝑣𝑣2subscriptproduct𝑣subscriptΣ𝐹subscriptsuperscriptsubscriptΔ𝑣0𝑣subscriptproduct𝑣subscriptΣ𝐹superscriptsubscriptsubscript𝑚𝑣𝑣21=\prod_{v\in\Sigma_{F}}|4^{-1}\Delta|_{v}=\prod_{v\in\Sigma_{F}}|\Delta_{v}^{0}|_{v}|m_{v}|_{v}^{2}=\prod_{v\in\Sigma_{F}}|\Delta_{v}^{0}|_{v}\prod_{v\in\Sigma_{F}}|m_{v}|_{v}^{2},

vΣF|Δv0mv|v(z+1)/2=vΣF|Δv0|vz+14=N(𝔇Δ)z+14,\prod_{v\in\Sigma_{F}}|\Delta_{v}^{0}m_{v}|_{v}^{-(z+1)/2}=\prod_{v\in\Sigma_{F}}|\Delta_{v}^{0}|_{v}^{-\frac{z+1}{4}}={\operatorname{N}}({\mathfrak{D}}_{\Delta})^{\frac{z+1}{4}},

where 𝔇Δ=FvΣfinΔv0𝔬vsubscript𝔇Δ𝐹subscriptproduct𝑣subscriptΣfinsuperscriptsubscriptΔ𝑣0subscript𝔬𝑣{\mathfrak{D}}_{\Delta}=F\cap\prod_{v\in\Sigma_{\rm fin}}\Delta_{v}^{0}\mathfrak{o}_{v}. Using the relation N(𝔇Δ)=N(𝔡E/F)vΣsplitΔDΣ|2|v2Nsubscript𝔇ΔNsubscript𝔡𝐸𝐹subscriptproduct𝑣superscriptsubscriptΣsplitΔ𝐷subscriptΣsuperscriptsubscript2𝑣2{\operatorname{N}}({\mathfrak{D}}_{\Delta})={\operatorname{N}}({\mathfrak{d}}_{E/F})\prod_{v\in\Sigma_{\rm split}^{\Delta}\cup D\cup\Sigma_{\infty}}|2|_{v}^{-2}, we see that the factor (7.8) is simplified in the desired form. ∎

Lemma 7.8.

For M>1𝑀1M>1 and ϵ>0italic-ϵ0\epsilon>0,

|(z21)ΛF(z+1)EΔ(z;12)|M,ϵ{vΣF|mv|v1}N(𝔇Δ)1+ϵ4+ϱ(z)4\displaystyle|(z^{2}-1)\Lambda_{F}(z+1)\,E^{\Delta}(z;1_{2})|\ll_{M,\epsilon}\{\prod_{v\in\Sigma_{F}}|m_{v}|_{v}^{-1}\}\,{\operatorname{N}}({\mathfrak{D}}_{\Delta})^{\frac{1+\epsilon}{4}+\frac{\varrho(z)}{4}}

uniformly in ΔF×(F×)2Δsuperscript𝐹superscriptsuperscript𝐹2\Delta\in F^{\times}-(F^{\times})^{2} and z𝑧z\in{\mathbb{C}} such that Re(z)[M,M]Re𝑧𝑀𝑀\operatorname{Re}(z)\in[-M,M], where ϱ(z)=max(|Re(z)|,1)italic-ϱ𝑧Re𝑧1\varrho(z)=\max(|\operatorname{Re}(z)|,1).

Proof.

The function (z21)ζF(z+12)superscript𝑧21subscript𝜁𝐹𝑧12(z^{2}-1)\zeta_{F}(\frac{z+1}{2}) is holomorphic and vertically bounded on {\mathbb{C}}. By two estimates |L(z+12,εΔ)|M,ϵN(𝔇Δ)ϵ|L(\frac{z+1}{2},\varepsilon_{\Delta})|\ll_{M,\epsilon}{\operatorname{N}}({\mathfrak{D}}_{\Delta})^{\epsilon} for Re(z)[1,M]Re𝑧1𝑀\operatorname{Re}(z)\in[1,M] and |L(z+12,εΔ)|M,ϵN(𝔇Δ)Re(z)2+ϵ|L(\frac{z+1}{2},\varepsilon_{\Delta})|\ll_{M,\epsilon}{\operatorname{N}}({\mathfrak{D}}_{\Delta})^{-\frac{\operatorname{Re}(z)}{2}+\epsilon} for Re(z)[M,1]Re𝑧𝑀1\operatorname{Re}(z)\in[-M,-1], a well-known argument by the Phragmen-Lindelöf principle yields a bound |L(z+12,εΔ)|ϵ{N(𝔇Δ)}14+ϵRe(z)4subscriptmuch-less-thanitalic-ϵ𝐿𝑧12subscript𝜀ΔsuperscriptNsubscript𝔇Δ14italic-ϵRe𝑧4|L(\tfrac{z+1}{2},\varepsilon_{\Delta})|\ll_{\epsilon}\{{\operatorname{N}}({\mathfrak{D}}_{\Delta})\}^{\frac{1}{4}+\epsilon-\frac{\operatorname{Re}(z)}{4}} uniformly valid for Re(z)[1,1]Re𝑧11\operatorname{Re}(z)\in[-1,1] and for any non-square ΔΔ\Delta. From this and Proposition 7.7, we have the desired bound easily. ∎

7.3. Explicit formulas of local orbital integrals

For vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}, let Φv(gv)subscriptΦ𝑣subscript𝑔𝑣\Phi_{v}(g_{v}) be the v𝑣v-component of the test function Φ(g)=Φl(𝔫|𝐬,g)Φ𝑔superscriptΦ𝑙conditional𝔫𝐬𝑔\Phi(g)=\Phi^{l}({\mathfrak{n}}|{\mathbf{s}},g) on G𝔸subscript𝐺𝔸G_{\mathbb{A}}. For z𝑧z\in{\mathbb{C}}, set

(7.9) 𝔈v(z)(γ^v)=𝔗Δ,v\GvΦv(g1γ^vg)φvΔ,(z)(g)𝑑gsuperscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣subscript\subscript𝔗Δ𝑣subscript𝐺𝑣subscriptΦ𝑣superscript𝑔1subscript^𝛾𝑣𝑔superscriptsubscript𝜑𝑣Δ𝑧𝑔differential-d𝑔\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=\int_{{\mathfrak{T}}_{\Delta,v}\backslash G_{v}}\Phi_{v}\left(g^{-1}\hat{\gamma}_{v}g\right)\,\varphi_{v}^{\Delta,(z)}(g)\,{{d}}g

with γ^v=[t2mv1Δv0t2mv]subscript^𝛾𝑣delimited-[]𝑡2subscript𝑚𝑣1superscriptsubscriptΔ𝑣0𝑡2subscript𝑚𝑣\hat{\gamma}_{v}=\left[\begin{smallmatrix}{\frac{t}{2m_{v}}}&1\\ \Delta_{v}^{0}&{\frac{t}{2m_{v}}}\end{smallmatrix}\right] if Δv01superscriptsubscriptΔ𝑣01\Delta_{v}^{0}\not=1 and γ^v=[t2mv+100t2mv1]subscript^𝛾𝑣delimited-[]𝑡2subscript𝑚𝑣100𝑡2subscript𝑚𝑣1\hat{\gamma}_{v}=\left[\begin{smallmatrix}{\frac{t}{2m_{v}}+1}&0\\ 0&{\frac{t}{2m_{v}}-1}\end{smallmatrix}\right] if Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1.

Theorem 7.9.
  • (1)

    Suppose vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Then for |Re(z)|<2lv1Re𝑧2subscript𝑙𝑣1|\operatorname{Re}(z)|<2l_{v}-1, we have

    𝔈v(z)(γ^v)=2|mv|v×𝒪vsgn(Δv0),(z)(t2mv).superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣2subscriptsubscript𝑚𝑣𝑣superscriptsubscript𝒪𝑣sgnsuperscriptsubscriptΔ𝑣0𝑧𝑡2subscript𝑚𝑣\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=2|m_{v}|_{v}\times{\mathcal{O}}_{v}^{{\rm{sgn}}(\Delta_{v}^{0}),(z)}\left(\tfrac{t}{2m_{v}}\right).
  • (2)

    Suppose vΣfin(SS(𝔫)Σdyadic)𝑣subscriptΣfin𝑆𝑆𝔫subscriptΣdyadicv\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}})\cup\Sigma_{\rm dyadic}) or vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} with Δv05superscriptsubscriptΔ𝑣05\Delta_{v}^{0}\neq 5. Then we have

    𝔈v(z)(γ^v)=qvdv/2|2mv|v𝒪v,0Δv0,(z)(n4mv2)δ(t2mv𝔬v×orn4mv2𝔭v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣2subscript2subscript𝑚𝑣𝑣superscriptsubscript𝒪𝑣0superscriptsubscriptΔ𝑣0𝑧𝑛4superscriptsubscript𝑚𝑣2𝛿𝑡2subscript𝑚𝑣superscriptsubscript𝔬𝑣or𝑛4superscriptsubscript𝑚𝑣2subscript𝔭𝑣\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=q_{v}^{-d_{v}/2}|2m_{v}|_{v}\,{\mathcal{O}}_{v,0}^{\Delta_{v}^{0},(z)}\left(\tfrac{n}{4m_{v}^{2}}\right)\,\delta\left(\tfrac{t}{2m_{v}}\not\in\mathfrak{o}_{v}^{\times}\,{\text{or}}\,\tfrac{n}{4m_{v}^{2}}\not\in{\mathfrak{p}}_{v}\right)

    if Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1, and otherwise

    𝔈v(z)(γ^v)=qvdv/2|mv|v𝒪v,0Δv0,(z)(nmv2){1(vΣdyadic,Δv0𝔬v×(𝔬v×)2,),δ(t2mv𝔬v×)(vΣdyadic,Δv0{1,5}),δ(t2mv𝔭v)(Δv0𝔭v𝔭v2).\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=q_{v}^{-d_{v}/2}|m_{v}|_{v}\,{\mathcal{O}}_{v,0}^{\Delta_{v}^{0},(z)}\left(\tfrac{n}{m_{v}^{2}}\right)\begin{cases}1\quad&(v\not\in\Sigma_{\rm dyadic},\,\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2},),\\ \delta\left(\tfrac{t}{2m_{v}}\not\in\mathfrak{o}_{v}^{\times}\right)\quad&(v\in\Sigma_{\rm dyadic},\,\Delta_{v}^{0}\in\{-1,-5\}),\\ \delta\left(\tfrac{t}{2m_{v}}\not\in{\mathfrak{p}}_{v}\right)\quad&(\Delta_{v}^{0}\in{\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2}).\end{cases}
  • (3)

    Suppose vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} with Δv0=5superscriptsubscriptΔ𝑣05\Delta_{v}^{0}=5. Then

    𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) =qvdv/2|2mv|v 2z12 3(1+2z)1𝒪v(n4mv2).absentsuperscriptsubscript𝑞𝑣subscript𝑑𝑣2subscript2subscript𝑚𝑣𝑣superscript2𝑧123superscript1superscript2𝑧1subscript𝒪𝑣𝑛4superscriptsubscript𝑚𝑣2\displaystyle=q_{v}^{-d_{v}/2}|2m_{v}|_{v}\,2^{\frac{-z-1}{2}}\,3(1+2^{-z})^{-1}\,{\mathcal{O}}_{v}\left(\tfrac{n}{4m_{v}^{2}}\right).
  • (4)

    Suppose vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}). Then

    𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) =qvdv/2|mv|v𝒪v,1Δv0,(z)(nmv2)×{δ(t2mv𝔬v×ornmv2𝔭v)(Δv0=1),δ(nmv2𝔬v)(Δv0𝔬v×(𝔬v×)2),δ(nmv2𝔭v)(Δv0𝔭v𝔭v2).absentsuperscriptsubscript𝑞𝑣subscript𝑑𝑣2subscriptsubscript𝑚𝑣𝑣superscriptsubscript𝒪𝑣1superscriptsubscriptΔ𝑣0𝑧𝑛superscriptsubscript𝑚𝑣2cases𝛿𝑡2subscript𝑚𝑣superscriptsubscript𝔬𝑣or𝑛superscriptsubscript𝑚𝑣2subscript𝔭𝑣superscriptsubscriptΔ𝑣01otherwise𝛿𝑛superscriptsubscript𝑚𝑣2subscript𝔬𝑣superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2otherwise𝛿𝑛superscriptsubscript𝑚𝑣2subscript𝔭𝑣superscriptsubscriptΔ𝑣0subscript𝔭𝑣superscriptsubscript𝔭𝑣2otherwise\displaystyle=q_{v}^{-d_{v}/2}|m_{v}|_{v}\,{\mathcal{O}}_{v,1}^{\Delta_{v}^{0},(z)}\left(\tfrac{n}{m_{v}^{2}}\right)\times\begin{cases}\delta\left(\tfrac{t}{2m_{v}}\not\in\mathfrak{o}_{v}^{\times}\,\text{or}\,\tfrac{n}{m_{v}^{2}}\not\in{\mathfrak{p}}_{v}\right)\quad(\Delta_{v}^{0}=1),\\ \delta\left(\tfrac{n}{m_{v}^{2}}\not\in\mathfrak{o}_{v}\right)\quad(\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}),\\ \delta\left(\tfrac{n}{m_{v}^{2}}\not\in{\mathfrak{p}}_{v}\right)\quad(\Delta_{v}^{0}\in{\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2}).\end{cases}
  • (5)

    Suppose vS𝑣𝑆v\in S. Then for Re(sv)>(|Re(z)|1)/2Resubscript𝑠𝑣Re𝑧12\operatorname{Re}(s_{v})>(|\operatorname{Re}(z)|-1)/2, we have

    𝔈v(z)(γ^v)=qvdv/2|mv|v𝒮vΔv0,(z)(sv;nmv2).superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣superscriptsubscript𝑞𝑣subscript𝑑𝑣2subscriptsubscript𝑚𝑣𝑣superscriptsubscript𝒮𝑣superscriptsubscriptΔ𝑣0𝑧subscript𝑠𝑣𝑛superscriptsubscript𝑚𝑣2\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=q_{v}^{-d_{v}/2}|m_{v}|_{v}\,{\mathcal{S}}_{v}^{\Delta_{v}^{0},(z)}\left(s_{v};\tfrac{n}{m_{v}^{2}}\right).
Proof.

The case Δv01superscriptsubscriptΔ𝑣01\Delta_{v}^{0}\not=1 will be treated in § 10. Suppose Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1. Since γ~𝒬FIrr~𝛾superscriptsubscript𝒬𝐹Irr\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm{Irr}}, we have n0𝑛0n\not=0, which in turn implies t2mv𝑡2subscript𝑚𝑣t\not=2m_{v}. Recall 𝔗Δ,v=Hvsubscript𝔗Δ𝑣subscript𝐻𝑣\mathfrak{T}_{\Delta,v}=H_{v}. Set a=t2mvFv{1}𝑎𝑡2subscript𝑚𝑣subscript𝐹𝑣1a=\frac{t}{2m_{v}}\in F_{v}-\{1\} and b=a+1a1𝑏𝑎1𝑎1b=\frac{a+1}{a-1}. Then, by adjusting the measures, we have

𝔈v(z)(γ^v)=superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣absent\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})= |2mv|vHv\GvΦv(g1[b001]g)φv(0,z)(g)𝑑gsubscript2subscript𝑚𝑣𝑣subscript\subscript𝐻𝑣subscript𝐺𝑣subscriptΦ𝑣superscript𝑔1delimited-[]𝑏001𝑔superscriptsubscript𝜑𝑣0𝑧𝑔differential-d𝑔\displaystyle|2m_{v}|_{v}\textstyle{\int}_{H_{v}\backslash G_{v}}\Phi_{v}\left(g^{-1}\left[\begin{smallmatrix}b&0\\ 0&1\end{smallmatrix}\right]g\right)\,\varphi_{v}^{(0,z)}(g)\,{{d}}g
=\displaystyle= |2mv|vFv(𝐊vΦv(k1[b(b1)x01]k)𝑑k)φv(0,z)([1x01])𝑑x=|2mv|v𝔉v(z)(b),subscript2subscript𝑚𝑣𝑣subscriptsubscript𝐹𝑣subscriptsubscript𝐊𝑣subscriptΦ𝑣superscript𝑘1delimited-[]𝑏𝑏1𝑥01𝑘differential-d𝑘superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥subscript2subscript𝑚𝑣𝑣superscriptsubscript𝔉𝑣𝑧𝑏\displaystyle|2m_{v}|_{v}\textstyle{\int}_{F_{v}}\left(\int_{{\mathbf{K}}_{v}}\Phi_{v}\left(k^{-1}\left[\begin{smallmatrix}b&(b-1)x\\ 0&1\end{smallmatrix}\right]k\right)\,{{d}}k\,\right)\varphi_{v}^{(0,z)}(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right])\,{{d}}x=|2m_{v}|_{v}\,{\mathfrak{F}}_{v}^{(z)}(b),

where 𝔉v(z)(b)superscriptsubscript𝔉𝑣𝑧𝑏{\mathfrak{F}}_{v}^{(z)}(b) is the integral treated in § 6.3. The case Δv0=+1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=+1 of (1) follows from Theorem 6.4 (1). The first case of (2) follows from Theorem 6.4 (2) if one notes the relation δ(a+1a1𝔬v×)=δ(a𝔬v×,a214𝔭v)𝛿𝑎1𝑎1superscriptsubscript𝔬𝑣𝛿formulae-sequence𝑎superscriptsubscript𝔬𝑣superscript𝑎214subscript𝔭𝑣\delta(\tfrac{a+1}{a-1}\not\in\mathfrak{o}_{v}^{\times})=\delta(a\in\mathfrak{o}_{v}^{\times},a^{2}-1\in 4{\mathfrak{p}}_{v}) by the identity |a+1a11|v2=|4a21|vsuperscriptsubscript𝑎1𝑎11𝑣2subscript4superscript𝑎21𝑣|\tfrac{a+1}{a-1}-1|_{v}^{2}=|\tfrac{4}{a^{2}-1}|_{v} for a𝑎a with |a1|v=|a+1|vsubscript𝑎1𝑣subscript𝑎1𝑣|a-1|_{v}=|a+1|_{v}, which is seen to be valid as |a+1a11|2=|4|v|a1|v2=|4|v|a1|v|a+1|v=|4|v|a21|vsuperscript𝑎1𝑎112subscript4𝑣superscriptsubscript𝑎1𝑣2subscript4𝑣subscript𝑎1𝑣subscript𝑎1𝑣subscript4𝑣subscriptsuperscript𝑎21𝑣|\tfrac{a+1}{a-1}-1|^{2}=\tfrac{|4|_{v}}{|a-1|_{v}^{2}}=\tfrac{|4|_{v}}{|a-1|_{v}|a+1|_{v}}=\tfrac{|4|_{v}}{|a^{2}-1|_{v}}. The first case of (4) follows from Theorem 6.4 (3) similarly. The case Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1 in (5) follows from Theorem 6.4 (4) by |a+1a11|v=|2|v|a1|v=|a|v1=|a21|1/2subscript𝑎1𝑎11𝑣subscript2𝑣subscript𝑎1𝑣superscriptsubscript𝑎𝑣1superscriptsuperscript𝑎2112|\tfrac{a+1}{a-1}-1|_{v}=\frac{|2|_{v}}{|a-1|_{v}}=|a|_{v}^{-1}=|a^{2}-1|^{-1/2} if |a|v>1subscript𝑎𝑣1|a|_{v}>1 and by |a+1a11|v2|a+1a1|v=|a21|vsuperscriptsubscript𝑎1𝑎11𝑣2subscript𝑎1𝑎1𝑣subscriptsuperscript𝑎21𝑣|\frac{a+1}{a-1}-1|_{v}^{-2}|\tfrac{a+1}{a-1}|_{v}=|a^{2}-1|_{v} if |a|v1subscript𝑎𝑣1|a|_{v}\leqslant 1. ∎

Lemma 7.10.

Let vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. Then

|t2mv|v=|Δv0t2t24n|v1/2,|nmv2|v=|Δv04nt24n|v=|(t2mv)2Δv0|v.formulae-sequencesubscript𝑡2subscript𝑚𝑣𝑣superscriptsubscriptsuperscriptsubscriptΔ𝑣0superscript𝑡2superscript𝑡24𝑛𝑣12subscript𝑛superscriptsubscript𝑚𝑣2𝑣subscriptsuperscriptsubscriptΔ𝑣04𝑛superscript𝑡24𝑛𝑣subscriptsuperscript𝑡2subscript𝑚𝑣2superscriptsubscriptΔ𝑣0𝑣\displaystyle|\tfrac{t}{2m_{v}}|_{v}=|\Delta_{v}^{0}\tfrac{t^{2}}{t^{2}-4n}|_{v}^{1/2},\quad|\tfrac{n}{m_{v}^{2}}|_{v}=|\Delta_{v}^{0}\tfrac{4n}{t^{2}-4n}|_{v}=|(\tfrac{t}{2m_{v}})^{2}-\Delta_{v}^{0}|_{v}.

If |t2mv|v<1subscript𝑡2subscript𝑚𝑣𝑣1|\tfrac{t}{2m_{v}}|_{v}<1, then |nmv2|v=|Δv0|vsubscript𝑛superscriptsubscript𝑚𝑣2𝑣subscriptsuperscriptsubscriptΔ𝑣0𝑣|\tfrac{n}{m_{v}^{2}}|_{v}=|\Delta_{v}^{0}|_{v}. If |t2mv|v=1subscript𝑡2subscript𝑚𝑣𝑣1|\tfrac{t}{2m_{v}}|_{v}=1, then

|nmv2|v={|t|v2|4n|v(Δv0=1,|t|v2>|4n|v),1(Δv01orΔv0=1,|t|v2|4n|v).|\tfrac{n}{m_{v}^{2}}|_{v}=\begin{cases}|t|_{v}^{-2}|4n|_{v}\quad(\Delta_{v}^{0}=1,\,|t|_{v}^{2}>|4n|_{v}),\\ 1\quad(\Delta_{v}^{0}\not=1\,\text{or}\,\Delta_{v}^{0}=1,\,|t|_{v}^{2}\leqslant|4n|_{v}).\end{cases}

except when Δv0𝔬v×(𝔬v×)2superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2} and v𝑣v is dyadic. If |t2mv|v>1subscript𝑡2subscript𝑚𝑣𝑣1|\tfrac{t}{2m_{v}}|_{v}>1, then |nmv2|v=|4nΔv0t24n|vsubscript𝑛superscriptsubscript𝑚𝑣2𝑣subscript4𝑛superscriptsubscriptΔ𝑣0superscript𝑡24𝑛𝑣|\tfrac{n}{m_{v}^{2}}|_{v}=|\frac{4n\Delta_{v}^{0}}{t^{2}-4n}|_{v}.

Proof.

This follows from the relation t24n=Δv0(2mv)2superscript𝑡24𝑛superscriptsubscriptΔ𝑣0superscript2subscript𝑚𝑣2t^{2}-4n=\Delta_{v}^{0}(2m_{v})^{2}. ∎

For vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} and m𝑚m\in{\mathbb{N}}, we set Uv(m)=1+𝔭vmsubscript𝑈𝑣𝑚1superscriptsubscript𝔭𝑣𝑚U_{v}(m)=1+{\mathfrak{p}}_{v}^{m}.

Lemma 7.11.

Let vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}.

  • (1)

    Suppose vΣdyadic𝑣subscriptΣdyadicv\not\in\Sigma_{\rm dyadic}, Δv0𝔬v×(𝔬v×)2superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}. Then

    |t2mv|v<1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}<1\, |t|v2<|4n|v,absentsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣\displaystyle\Longleftrightarrow\,|t|_{v}^{2}<|4n|_{v},
    |t2mv|v=1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}=1\, t0,4nt2𝔬v×Uv(1),absentformulae-sequence𝑡04𝑛superscript𝑡2superscriptsubscript𝔬𝑣subscript𝑈𝑣1\displaystyle\Longleftrightarrow\,t\not=0,\,\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}-U_{v}(1),
    |t2mv|v>1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}>1\, t0,4nt2Uv(1).absentformulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣1\displaystyle\Longleftrightarrow\,t\not=0,\,\tfrac{4n}{t^{2}}\in U_{v}(1).
  • (2)

    Suppose Δv0𝔭v𝔭v2superscriptsubscriptΔ𝑣0subscript𝔭𝑣superscriptsubscript𝔭𝑣2\Delta_{v}^{0}\in{\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2}. Then t0,4nt2𝔬v×Uv(1)formulae-sequence𝑡04𝑛superscript𝑡2superscriptsubscript𝔬𝑣subscript𝑈𝑣1t\not=0,\,\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}-U_{v}(1) never happens. We have

    |t2mv|v<1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}<1\, |t|v2<|4n|v,absentsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣\displaystyle\Longleftrightarrow\,|t|_{v}^{2}<|4n|_{v},
    |t2mv|v=1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}=1\, t0,4nt2Uv(1)Uv(2),absentformulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣1subscript𝑈𝑣2\displaystyle\Longleftrightarrow\,t\not=0,\,\tfrac{4n}{t^{2}}\in U_{v}(1)-U_{v}(2),
    |t2mv|v>1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}>1\, t0,4nt2Uv(2).absentformulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣2\displaystyle\Longleftrightarrow\,t\not=0,\,\tfrac{4n}{t^{2}}\in U_{v}(2).
  • (3)

    Suppose Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1. Then

    |t2mv|v<1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}<1\, |t|v2<|4n|v,absentsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣\displaystyle\Longleftrightarrow\,|t|_{v}^{2}<|4n|_{v},
    |t2mv|v=1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}=1\, |t|v2>|4n|vort0,4nt2𝔬v×Uv(1),absentformulae-sequencesuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣orformulae-sequence𝑡04𝑛superscript𝑡2superscriptsubscript𝔬𝑣subscript𝑈𝑣1\displaystyle\Longleftrightarrow\,|t|_{v}^{2}>|4n|_{v}\quad\text{or}\quad t\not=0,\,\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}-U_{v}(1),
    |t2mv|v>1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}>1\, t0,4nt2Uv(1).absentformulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣1\displaystyle\Longleftrightarrow\,t\not=0,\,\tfrac{4n}{t^{2}}\in U_{v}(1).
  • (4)

    Suppose Fv=2subscript𝐹𝑣subscript2F_{v}={\mathbb{Q}}_{2} and Δv0=5superscriptsubscriptΔ𝑣05\Delta_{v}^{0}=5. Then

    |t2mv|v<1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}<1\, |t|v2<|4n|v,absentsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣\displaystyle\Longleftrightarrow\,|t|_{v}^{2}<|4n|_{v},
    |t2mv|v=1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}=1\, t0,nt2𝔬v×,absentformulae-sequence𝑡0𝑛superscript𝑡2superscriptsubscript𝔬𝑣\displaystyle\Longleftrightarrow\,t\not=0,\,\tfrac{n}{t^{2}}\in\mathfrak{o}_{v}^{\times},
    |t2mv|v>1subscript𝑡2subscript𝑚𝑣𝑣1\displaystyle|\tfrac{t}{2m_{v}}|_{v}>1\, t0,4nt2𝔬v×.absentformulae-sequence𝑡04𝑛superscript𝑡2superscriptsubscript𝔬𝑣\displaystyle\Longleftrightarrow\,t\not=0,\,\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}.

    Suppose Fv=2subscript𝐹𝑣subscript2F_{v}={\mathbb{Q}}_{2} and Δv0{1,5}superscriptsubscriptΔ𝑣015\Delta_{v}^{0}\in\{-1,-5\}. Then the above equivalences hold if we replace the n𝑛n in the second equivalence with 2n2𝑛2n.

Proof.

Since Δ(F×)2Δsuperscriptsuperscript𝐹2\Delta\not\in(F^{\times})^{2}, we have n0𝑛0n\not=0; thus |t|v2|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}\geqslant|4n|_{v} implies tn0𝑡𝑛0tn\not=0. We prove the case (1). From t24n=Δv0(2mv)2superscript𝑡24𝑛superscriptsubscriptΔ𝑣0superscript2subscript𝑚𝑣2t^{2}-4n=\Delta_{v}^{0}(2m_{v})^{2}, we have |(t2mv)2Δv0|v=|nmv2|vsubscriptsuperscript𝑡2subscript𝑚𝑣2superscriptsubscriptΔ𝑣0𝑣subscript𝑛superscriptsubscript𝑚𝑣2𝑣|(\tfrac{t}{2m_{v}})^{2}-\Delta_{v}^{0}|_{v}=|\tfrac{n}{m_{v}^{2}}|_{v} and |2mv|v2=|t24n|vsuperscriptsubscript2subscript𝑚𝑣𝑣2subscriptsuperscript𝑡24𝑛𝑣|2m_{v}|_{v}^{2}=|t^{2}-4n|_{v}. Assume |t2mv|v1subscript𝑡2subscript𝑚𝑣𝑣1|\frac{t}{2m_{v}}|_{v}\leqslant 1. Then |nmv2|v=|(t2mv)2Δv0|v=1subscript𝑛superscriptsubscript𝑚𝑣2𝑣subscriptsuperscript𝑡2subscript𝑚𝑣2superscriptsubscriptΔ𝑣0𝑣1|\frac{n}{m_{v}^{2}}|_{v}=|(\frac{t}{2m_{v}})^{2}-\Delta_{v}^{0}|_{v}=1 by Lemma 7.1. Hence |4n|v=|2mv|v2=|t24n|vsubscript4𝑛𝑣superscriptsubscript2subscript𝑚𝑣𝑣2subscriptsuperscript𝑡24𝑛𝑣|4n|_{v}=|2m_{v}|_{v}^{2}=|t^{2}-4n|_{v}, which yields |t|v2|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}\leqslant|4n|_{v}. If |t|v2=|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}=|4n|_{v}, then |t|v2|2mv|v2=|t24n|vsuperscriptsubscript𝑡𝑣2superscriptsubscript2subscript𝑚𝑣𝑣2subscriptsuperscript𝑡24𝑛𝑣|t|_{v}^{2}\leqslant|2m_{v}|_{v}^{2}=|t^{2}-4n|_{v} gives us |14nt2|v=1subscript14𝑛superscript𝑡2𝑣1|1-\frac{4n}{t^{2}}|_{v}=1, or equivalently 4nt2𝔬v×Uv(1)4𝑛superscript𝑡2superscriptsubscript𝔬𝑣subscript𝑈𝑣1\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}-U_{v}(1), which in turns implies |t2mv|v=1subscript𝑡2subscript𝑚𝑣𝑣1|\tfrac{t}{2m_{v}}|_{v}=1 as |4mv2|v=|t|v2|14nt2|v=|t|v2subscript4superscriptsubscript𝑚𝑣2𝑣superscriptsubscript𝑡𝑣2subscript14𝑛superscript𝑡2𝑣superscriptsubscript𝑡𝑣2|4m_{v}^{2}|_{v}=|t|_{v}^{2}|1-\tfrac{4n}{t^{2}}|_{v}=|t|_{v}^{2}. The condition |t|v2<|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}<|4n|_{v} implies |t|v2<|t24n|v=|4mv2|vsuperscriptsubscript𝑡𝑣2subscriptsuperscript𝑡24𝑛𝑣subscript4superscriptsubscript𝑚𝑣2𝑣|t|_{v}^{2}<|t^{2}-4n|_{v}=|4m_{v}^{2}|_{v}. This completes the first two cases of (1).

Assume |t2mv|v>1subscript𝑡2subscript𝑚𝑣𝑣1|\frac{t}{2m_{v}}|_{v}>1. Then |t|v2>|4mv2|v=|t24n|vsuperscriptsubscript𝑡𝑣2subscript4superscriptsubscript𝑚𝑣2𝑣subscriptsuperscript𝑡24𝑛𝑣|t|_{v}^{2}>|4m_{v}^{2}|_{v}=|t^{2}-4n|_{v}, which yields |t|v2|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}\leqslant|4n|_{v}. From this remark, we obtain the third case of (1) from the first two cases.

The remaining assertions are proved similarly. ∎

The following proposition combined with Theorem 7.9 gives us a convenient description of the local orbital integrals 𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) in terms of (t:n)F(t:n)_{F}.

Proposition 7.12.

Let (t:n)F𝒬FIrr(t:n)_{F}\in{\mathcal{Q}}^{\rm Irr}_{F} and vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin}. Write t24n=(2mv)2Δv0superscript𝑡24𝑛superscript2subscript𝑚𝑣2superscriptsubscriptΔ𝑣0t^{2}-4n=(2m_{v})^{2}\Delta_{v}^{0} with mvFvsubscript𝑚𝑣subscript𝐹𝑣m_{v}\in F_{v}, Δv0(𝔭v𝔭v2)(𝔬v×(𝔬v×)2){1}superscriptsubscriptΔ𝑣0subscript𝔭𝑣superscriptsubscript𝔭𝑣2superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣21\Delta_{v}^{0}\in({\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2})\cup(\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2})\cup\{1\}. Then

  • (1)

    Suppose ordv(t24n)2subscriptord𝑣superscript𝑡24𝑛2\operatorname{ord}_{v}(t^{2}-4n)\in 2{\mathbb{Z}} and t24n(Fv×)2superscript𝑡24𝑛superscriptsuperscriptsubscript𝐹𝑣2t^{2}-4n\not\in(F_{v}^{\times})^{2}. If vΣdyadic𝑣subscriptΣdyadicv\not\in\Sigma_{\rm dyadic}, then |t|v2|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}\leqslant|4n|_{v} and

    |nmv2|v={|4nt24n|v(t0,4nt2Uv(1)),1(otherwise).subscript𝑛superscriptsubscript𝑚𝑣2𝑣casessubscript4𝑛superscript𝑡24𝑛𝑣formulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣11otherwise\displaystyle|\tfrac{n}{m_{v}^{2}}|_{v}=\begin{cases}|\tfrac{4n}{t^{2}-4n}|_{v}\quad&(t\not=0,\tfrac{4n}{t^{2}}\in U_{v}(1)),\\ 1\quad&(\text{otherwise}).\end{cases}

    If Fv2subscript𝐹𝑣subscript2F_{v}\cong{\mathbb{Q}}_{2} and Δv0=5superscriptsubscriptΔ𝑣05\Delta_{v}^{0}=5, then |t|v2|n|vsuperscriptsubscript𝑡𝑣2subscript𝑛𝑣|t|_{v}^{2}\leqslant|n|_{v} and

    |nmv2|v={1(|t|v2<|4n|vort0,nt2𝔬v×),|4nt24n|v(t0,4nt2𝔬v×).subscript𝑛superscriptsubscript𝑚𝑣2𝑣cases1formulae-sequencesuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣orformulae-sequence𝑡0𝑛superscript𝑡2superscriptsubscript𝔬𝑣subscript4𝑛superscript𝑡24𝑛𝑣formulae-sequence𝑡04𝑛superscript𝑡2superscriptsubscript𝔬𝑣\displaystyle|\tfrac{n}{m_{v}^{2}}|_{v}=\begin{cases}1\quad&(|t|_{v}^{2}<|4n|_{v}\quad\text{or}\quad t\not=0,\tfrac{n}{t^{2}}\in\mathfrak{o}_{v}^{\times}),\\ |\tfrac{4n}{t^{2}-4n}|_{v}\quad&(t\not=0,\,\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}).\end{cases}

    If Fv2subscript𝐹𝑣subscript2F_{v}\cong{\mathbb{Q}}_{2} and Δv0{1,5}superscriptsubscriptΔ𝑣015\Delta_{v}^{0}\in\{-1,-5\}, then |t|v2<|n|vsubscriptsuperscript𝑡2𝑣subscript𝑛𝑣|t|^{2}_{v}<|n|_{v} and

    |nmv2|v={1(|t|v2<|4n|v),21(t0,2nt2𝔬v×),|4nt24n|v(t0,4nt2𝔬v×).subscript𝑛superscriptsubscript𝑚𝑣2𝑣cases1superscriptsubscript𝑡𝑣2subscript4𝑛𝑣superscript21formulae-sequence𝑡02𝑛superscript𝑡2superscriptsubscript𝔬𝑣subscript4𝑛superscript𝑡24𝑛𝑣formulae-sequence𝑡04𝑛superscript𝑡2superscriptsubscript𝔬𝑣\displaystyle|\tfrac{n}{m_{v}^{2}}|_{v}=\begin{cases}1\quad&(|t|_{v}^{2}<|4n|_{v}),\\ 2^{-1}\quad&(t\not=0,\,\tfrac{2n}{t^{2}}\in\mathfrak{o}_{v}^{\times}),\\ |\tfrac{4n}{t^{2}-4n}|_{v}\quad&(t\not=0,\,\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}).\end{cases}
  • (2)

    If ordv(t24n)1+2subscriptord𝑣superscript𝑡24𝑛12\operatorname{ord}_{v}(t^{2}-4n)\in 1+2{\mathbb{Z}}. Then |t|v2|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}\leqslant|4n|_{v}, and

    |nmv2|v={qv1(|t|v2<|4n|v),1(t0,4nt2Uv(1)Uv(2)),qv1|4nt24n|v(t0,4nt2Uv(2)).subscript𝑛superscriptsubscript𝑚𝑣2𝑣casessuperscriptsubscript𝑞𝑣1superscriptsubscript𝑡𝑣2subscript4𝑛𝑣1formulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣1subscript𝑈𝑣2superscriptsubscript𝑞𝑣1subscript4𝑛superscript𝑡24𝑛𝑣formulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣2\displaystyle|\tfrac{n}{m_{v}^{2}}|_{v}=\begin{cases}q_{v}^{-1}\quad&(|t|_{v}^{2}<|4n|_{v}),\\ 1\quad&(t\not=0,\tfrac{4n}{t^{2}}\in U_{v}(1)-U_{v}(2)),\\ q_{v}^{-1}|\tfrac{4n}{t^{2}-4n}|_{v}\quad&(t\not=0,\tfrac{4n}{t^{2}}\in U_{v}(2)).\end{cases}
  • (3)

    If t24n(Fv×)2superscript𝑡24𝑛superscriptsuperscriptsubscript𝐹𝑣2t^{2}-4n\in(F_{v}^{\times})^{2}. Then

    |nmv2|v={1(|t|v2<|4n|vort0,4nt2𝔬v×Uv(1)),|t|v2|4n|v(|t|v2>|4n|v),|4nt24n|v(t0,4nt2Uv(1)).subscript𝑛superscriptsubscript𝑚𝑣2𝑣cases1formulae-sequencesuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣orformulae-sequence𝑡04𝑛superscript𝑡2superscriptsubscript𝔬𝑣subscript𝑈𝑣1superscriptsubscript𝑡𝑣2subscript4𝑛𝑣superscriptsubscript𝑡𝑣2subscript4𝑛𝑣subscript4𝑛superscript𝑡24𝑛𝑣formulae-sequence𝑡04𝑛superscript𝑡2subscript𝑈𝑣1\displaystyle|\tfrac{n}{m_{v}^{2}}|_{v}=\begin{cases}1\quad&(|t|_{v}^{2}<|4n|_{v}\quad\text{or}\quad t\not=0,\tfrac{4n}{t^{2}}\in\mathfrak{o}_{v}^{\times}-U_{v}(1)),\\ |t|_{v}^{-2}|4n|_{v}\quad&(|t|_{v}^{2}>|4n|_{v}),\\ |\tfrac{4n}{t^{2}-4n}|_{v}\quad&(t\not=0,\tfrac{4n}{t^{2}}\in U_{v}(1)).\end{cases}
Proof.

This follows from Lemma 7.10, 7.11 immediately. ∎

7.4. The absolute convergence of the F𝐹F-elliptic term

In this subsection, we prove the following.

Theorem 7.13.

For any sufficiently small ϵ>0italic-ϵ0\epsilon>0, we have

(7.10) γ=(t:n)F𝒬FIrr|(z21)ΛF(z+1)EΔ(z;12)|vΣF|𝔈v(z)(γ^v)|<+\displaystyle\sum_{\gamma=(t:n)_{F}\in{\mathcal{Q}}_{F}^{\rm{Irr}}}|(z^{2}-1)\Lambda_{F}(z+1)E^{\Delta}(z;1_{2})|\prod_{v\in\Sigma_{F}}|{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})|<+\infty

uniformly in (z,𝐬)×𝔛S𝑧𝐬subscript𝔛𝑆(z,{\mathbf{s}})\in{\mathbb{C}}\times{\mathfrak{X}}_{S} such that Re(z)[2l¯+3+ϵ,l¯1/2ϵ]Re𝑧2¯𝑙3italic-ϵ¯𝑙12italic-ϵ\operatorname{Re}(z)\in[-2\underline{l}+3+\epsilon,\underline{l}-1/2-\epsilon], minvSRe(sv)>2l¯+dF1subscript𝑣𝑆Resubscript𝑠𝑣2¯𝑙subscript𝑑𝐹1\min_{v\in S}\operatorname{Re}(s_{v})>2{\underline{l}}+d_{F}-1.

To prove this, we need estimates of local orbital integrals, which are given by the following lemmas.

Lemma 7.14.
  • (1)

    We have 𝒪v,0δ,(z)(a)=1superscriptsubscript𝒪𝑣0𝛿𝑧𝑎1{\mathcal{O}}_{v,0}^{\delta,(z)}(a)=1 if a𝔬v×𝑎superscriptsubscript𝔬𝑣a\in\mathfrak{o}_{v}^{\times}, and

    |𝒪v,0δ,(z)(a)|3|ordv(a)||a|v|Re(z)|+14,aFv𝔬v.formulae-sequencesuperscriptsubscript𝒪𝑣0𝛿𝑧𝑎3subscriptord𝑣𝑎superscriptsubscript𝑎𝑣Re𝑧14𝑎subscript𝐹𝑣subscript𝔬𝑣\displaystyle|{\mathcal{O}}_{v,0}^{\delta,(z)}(a)|\leqslant 3|\operatorname{ord}_{v}(a)|\,|a|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}},\quad a\in F_{v}-\mathfrak{o}_{v}.
  • (2)

    We have |𝒪v,1δ,(z)(a)|2(1+qv)1superscriptsubscript𝒪𝑣1𝛿𝑧𝑎2superscript1subscript𝑞𝑣1|{\mathcal{O}}_{v,1}^{\delta,(z)}(a)|\leqslant 2(1+q_{v})^{-1} if a𝔬v×𝑎superscriptsubscript𝔬𝑣a\in\mathfrak{o}_{v}^{\times}, and

    |𝒪v,1δ,(z)(a)|4(1+qv1/2)1+qv(1+|ordv(a)|)|a|v|Re(z)|+14,aFv𝔬v.formulae-sequencesuperscriptsubscript𝒪𝑣1𝛿𝑧𝑎41superscriptsubscript𝑞𝑣121subscript𝑞𝑣1subscriptord𝑣𝑎superscriptsubscript𝑎𝑣Re𝑧14𝑎subscript𝐹𝑣subscript𝔬𝑣\displaystyle|{\mathcal{O}}_{v,1}^{\delta,(z)}(a)|\leqslant\tfrac{4(1+q_{v}^{1/2})}{1+q_{v}}(1+|\operatorname{ord}_{v}(a)|)|a|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}},\quad a\in F_{v}-\mathfrak{o}_{v}.
  • (3)

    Suppose Re(s)>1Re𝑠1\operatorname{Re}(s)>1 and Re(s)>(|Re(z)|+1)/2Re𝑠Re𝑧12\operatorname{Re}(s)>(|\operatorname{Re}(z)|+1)/2. Then for any ϵ>0italic-ϵ0\epsilon>0, we have

    |𝒮vδ,(z)(s;a)|16{1+max(0,ordv(a))}max(1,|a|v1)Re(s)+|Re(z)|4+ϵ|a|v|Re(z)|+14+ϵ\displaystyle|{\mathcal{S}}_{v}^{\delta,(z)}(s;a)|\leqslant 16\{1+\max(0,-\operatorname{ord}_{v}(a))\}\max(1,|a|_{v}^{-1})^{\frac{-\operatorname{Re}(s)+|\operatorname{Re}(z)|}{4}+\epsilon}|a|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}+\epsilon}

    for any aFv×𝑎superscriptsubscript𝐹𝑣a\in F_{v}^{\times} with ordv(a)0(2)subscriptord𝑣𝑎subscript02\operatorname{ord}_{v}(a)\in\mathbb{N}_{0}\cup(-2\mathbb{N}).

  • (4)

    Let vΣ𝑣subscriptΣv\in\Sigma_{\infty}. For any 0<σ<2lv10𝜎2subscript𝑙𝑣10<\sigma<2l_{v}-1,

    |𝒪v+,(z)(a)|δ(|a|v>1)min(1,|a1|v)lv/2|a|v|Re(z)|+12,aFv×\displaystyle|{\mathcal{O}}_{v}^{+,(z)}(a)|\ll\delta(|a|_{v}>1)\,\min(1,|a-1|_{v})^{l_{v}/2}|a|_{v}^{\frac{|\operatorname{Re}(z)|+1}{2}},\quad a\in F_{v}^{\times}

    uniformly for z𝑧z\in{\mathbb{C}} on the strip |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma.

  • (5)

    Let vΣ𝑣subscriptΣv\in\Sigma_{\infty}. For any subinterval (σ1,σ2)(2lv+1,lv1/2)subscript𝜎1subscript𝜎22subscript𝑙𝑣1subscript𝑙𝑣12(\sigma_{1},\sigma_{2})\subset(-2l_{v}+1,l_{v}-1/2), we have

    |𝒪v,(z)(a)|(1+|a|v)12lv4,aFv×formulae-sequencemuch-less-thansuperscriptsubscript𝒪𝑣𝑧𝑎superscript1subscript𝑎𝑣12subscript𝑙𝑣4𝑎superscriptsubscript𝐹𝑣\displaystyle|{\mathcal{O}}_{v}^{-,(z)}(a)|\ll(1+|a|_{v})^{\frac{1-2l_{v}}{4}},\quad a\in F_{v}^{\times}

    uniformly for z𝑧z\in{\mathbb{C}} on the strip σ1Re(z)σ2subscript𝜎1Re𝑧subscript𝜎2\sigma_{1}\leqslant\operatorname{Re}(z)\leqslant\sigma_{2}.

Proof.

When δ(Fv×)2𝛿superscriptsuperscriptsubscript𝐹𝑣2\delta\in(F_{v}^{\times})^{2}, the estimates in (1), (2) and (3) follow from the proof of Lemma 6.5, Lemma 6.7 and Corollary 6.9, respectively. The proofs are modified to be applied to the case δFv×(Fv×)2𝛿subscriptsuperscript𝐹𝑣superscriptsuperscriptsubscript𝐹𝑣2\delta\in F^{\times}_{v}-(F_{v}^{\times})^{2}. The estimate (4) is given by Corollary 6.11. From §10.2.1, to prove (5) it suffices to estimate 0ylv+z12(y2+1±2ayi)lv1/2d×ysuperscriptsubscript0superscript𝑦subscript𝑙𝑣𝑧12superscriptplus-or-minussuperscript𝑦212𝑎𝑦𝑖subscript𝑙𝑣12superscript𝑑𝑦\int_{0}^{\infty}\frac{y^{l_{v}+\frac{z-1}{2}}}{(y^{2}+1\pm 2ayi)^{l_{v}-1/2}}d^{\times}y, which is majorized by the sum of

01ylv+Re(z)12(y2+1)lv1/2d×y1,1ylv+Re(z)12(4|a|vy3)lv/21/4d×y|a|v1/4lv/2.formulae-sequencemuch-less-thansuperscriptsubscript01superscript𝑦subscript𝑙𝑣Re𝑧12superscriptsuperscript𝑦21subscript𝑙𝑣12superscript𝑑𝑦1much-less-thansuperscriptsubscript1superscript𝑦subscript𝑙𝑣Re𝑧12superscript4subscript𝑎𝑣superscript𝑦3subscript𝑙𝑣214superscript𝑑𝑦superscriptsubscript𝑎𝑣14subscript𝑙𝑣2\int_{0}^{1}\frac{y^{l_{v}+\frac{\operatorname{Re}(z)-1}{2}}}{(y^{2}+1)^{l_{v}-1/2}}d^{\times}y\ll 1,\qquad\int_{1}^{\infty}\frac{y^{l_{v}+\frac{\operatorname{Re}(z)-1}{2}}}{(4|a|_{v}y^{3})^{l_{v}/2-1/4}}d^{\times}y\ll|a|_{v}^{1/4-l_{v}/2}.

To be precise, let aa(v)maps-to𝑎superscript𝑎𝑣a\mapsto a^{(v)} denote the natural embedding FFv𝐹subscript𝐹𝑣F\hookrightarrow F_{v}. As usual, ordv(a(v))subscriptord𝑣superscript𝑎𝑣\operatorname{ord}_{v}(a^{(v)}) and |a(v)|vsubscriptsuperscript𝑎𝑣𝑣|a^{(v)}|_{v} for aF×𝑎superscript𝐹a\in F^{\times} will be abbreviated to ordv(a)subscriptord𝑣𝑎\operatorname{ord}_{v}(a) and |a|vsubscript𝑎𝑣|a|_{v}, respectively.

Lemma 7.15.

Let γ=(t:n)F𝒬FIrr\gamma=(t:n)_{F}\in{\mathcal{Q}}_{F}^{\rm Irr} be such that t,n𝔬𝑡𝑛𝔬t,n\in\mathfrak{o} and

  • (a)

    For all vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}), 4n𝔬v+t𝔬v=𝔬v4𝑛subscript𝔬𝑣𝑡subscript𝔬𝑣subscript𝔬𝑣4n\mathfrak{o}_{v}+t\mathfrak{o}_{v}=\mathfrak{o}_{v},

  • (b)

    vS(𝔫)𝔈v(z)(γ^v)0subscriptproduct𝑣𝑆𝔫superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣0\prod_{v\in S({\mathfrak{n}})}{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})\not=0.

Then 4n4𝑛4n is relatively prime to 𝔫𝔫{\mathfrak{n}}. If we define a divisor 𝔫2subscript𝔫2{\mathfrak{n}}_{2} of 𝔫𝔫{\mathfrak{n}} by S(𝔫2)={vS(𝔫)|Δv0=1,|t|v=|n|v=|mv|v=1}𝑆subscript𝔫2conditional-set𝑣𝑆𝔫formulae-sequencesuperscriptsubscriptΔ𝑣01subscript𝑡𝑣subscript𝑛𝑣subscriptsubscript𝑚𝑣𝑣1S({\mathfrak{n}}_{2})=\{v\in S({\mathfrak{n}})|\Delta_{v}^{0}=1,\,|t|_{v}=|n|_{v}=|m_{v}|_{v}=1\} and set 𝔫1=𝔫𝔫21subscript𝔫1𝔫superscriptsubscript𝔫21{\mathfrak{n}}_{1}={\mathfrak{n}}{\mathfrak{n}}_{2}^{-1}, 𝔠=t𝔬+𝔫𝔠𝑡𝔬𝔫{\mathfrak{c}}=t\mathfrak{o}+{\mathfrak{n}} and 𝔠=𝔫1𝔠1superscript𝔠subscript𝔫1superscript𝔠1{\mathfrak{c}}^{\prime}={\mathfrak{n}}_{1}{\mathfrak{c}}^{-1}, then 𝔫1=𝔠𝔠subscript𝔫1𝔠superscript𝔠{\mathfrak{n}}_{1}={\mathfrak{c}}{\mathfrak{c}}^{\prime}, ((t24n)(v))vS(𝔠)vS(𝔠)(𝔬v×)2subscriptsuperscriptsuperscript𝑡24𝑛𝑣𝑣𝑆𝔠subscriptproduct𝑣𝑆𝔠superscriptsuperscriptsubscript𝔬𝑣2((t^{2}-4n)^{(v)})_{v\in S({\mathfrak{c}})}\in\prod_{v\in S({\mathfrak{c}})}(\mathfrak{o}_{v}^{\times})^{2} and (t24n)𝔬(𝔠)2superscript𝑡24𝑛𝔬superscriptsuperscript𝔠2(t^{2}-4n)\mathfrak{o}\subset({\mathfrak{c}}^{\prime})^{2}.

If n𝑛n further satisfies the condition

  • (c)

    n(mod𝔭v)annotated𝑛pmodsubscript𝔭𝑣n\pmod{{\mathfrak{p}}_{v}} is not a square residue for some vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}),

then t0𝑡0t\not=0.

Proof.

Let vS(𝔠)𝑣𝑆𝔠v\in S({\mathfrak{c}}). Then t(v)𝔭vsuperscript𝑡𝑣subscript𝔭𝑣t^{(v)}\in{\mathfrak{p}}_{v}. From (a), t(v)superscript𝑡𝑣t^{(v)} and (t24n)(v)superscriptsuperscript𝑡24𝑛𝑣(t^{2}-4n)^{(v)} are relatively prime in 𝔬vsubscript𝔬𝑣\mathfrak{o}_{v}. Hence (t24n)(v)𝔬v×superscriptsuperscript𝑡24𝑛𝑣superscriptsubscript𝔬𝑣(t^{2}-4n)^{(v)}\in\mathfrak{o}_{v}^{\times}. From the relation (t24n)(v)=4mv2Δv0superscriptsuperscript𝑡24𝑛𝑣4subscriptsuperscript𝑚2𝑣superscriptsubscriptΔ𝑣0(t^{2}-4n)^{(v)}=4m^{2}_{v}\Delta_{v}^{0} with ordv(Δv0){0,1}subscriptord𝑣superscriptsubscriptΔ𝑣001\operatorname{ord}_{v}(\Delta_{v}^{0})\in\{0,1\}, we conclude mv𝔬v×subscript𝑚𝑣superscriptsubscript𝔬𝑣m_{v}\in\mathfrak{o}_{v}^{\times} and Δv0𝔬v×superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}. If Δv0𝔬v×(𝔬v×)2superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}, from Theorem 7.9 (3) and (1.3), the non-vanishing of 𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) implies |nmv2|v>1subscript𝑛superscriptsubscript𝑚𝑣2𝑣1|\tfrac{n}{m_{v}^{2}}|_{v}>1. From Lemma 7.12 (1), we have |4n|v=|t|v2subscript4𝑛𝑣superscriptsubscript𝑡𝑣2|4n|_{v}=|t|_{v}^{2}, which causes a contradiction when combined with |t|v<1subscript𝑡𝑣1|t|_{v}<1 and |t24n|v=1subscriptsuperscript𝑡24𝑛𝑣1|t^{2}-4n|_{v}=1. Hence we must have Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1 and (t24n)(v)=(2mv)2(𝔬v×)2superscriptsuperscript𝑡24𝑛𝑣superscript2subscript𝑚𝑣2superscriptsuperscriptsubscript𝔬𝑣2(t^{2}-4n)^{(v)}=(2m_{v})^{2}\in(\mathfrak{o}_{v}^{\times})^{2} as desired. Let vS(𝔠)𝑣𝑆superscript𝔠v\in S({\mathfrak{c}}^{\prime}). Then t(v)𝔬v×superscript𝑡𝑣superscriptsubscript𝔬𝑣t^{(v)}\in\mathfrak{o}_{v}^{\times}. Hence |t|v=1subscript𝑡𝑣1|t|_{v}=1 and |4n|v1subscript4𝑛𝑣1|4n|_{v}\leqslant 1. From Theorem 7.9, Lemma 7.11 and Proposition 7.12, the non-vanishing of 𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) implies that 4nt2Uv(1)4𝑛superscript𝑡2subscript𝑈𝑣1\tfrac{4n}{t^{2}}\in U_{v}(1) if ordv(t24n)2subscriptord𝑣superscript𝑡24𝑛2\operatorname{ord}_{v}(t^{2}-4n)\in 2{\mathbb{Z}} and 4nt2Uv(2)4𝑛superscript𝑡2subscript𝑈𝑣2\tfrac{4n}{t^{2}}\in U_{v}(2) if ordv(t24n)1+2subscriptord𝑣superscript𝑡24𝑛12\operatorname{ord}_{v}(t^{2}-4n)\in 1+2{\mathbb{Z}}. Therefore, (t24n)(v)𝔭v2superscriptsuperscript𝑡24𝑛𝑣superscriptsubscript𝔭𝑣2(t^{2}-4n)^{(v)}\in{\mathfrak{p}}_{v}^{2} and |4n|v=|t|v2=1subscript4𝑛𝑣superscriptsubscript𝑡𝑣21|4n|_{v}=|t|_{v}^{2}=1 for all vS(𝔠)𝑣𝑆superscript𝔠v\in S({\mathfrak{c}}^{\prime}) as desired. Suppose t=0𝑡0t=0. Then t24n=4nsuperscript𝑡24𝑛4𝑛t^{2}-4n=-4n is prime to 𝔫𝔫{\mathfrak{n}} as shown above. Hence for all vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}), we have mv,Δv0𝔬v×subscript𝑚𝑣superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣m_{v},\,\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times} and |nmv2|v=1subscript𝑛superscriptsubscript𝑚𝑣2𝑣1|\tfrac{n}{m_{v}^{2}}|_{v}=1. From (c), there is vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}) such that Δv0𝔬v×(𝔬v×)2superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}. Hence from Theorem 7.9 (4), we obtain 𝔈v(z)(γ^v)=0superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣0{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=0 a contradiction. This shows t0𝑡0t\not=0. ∎

Let 𝒬FSsuperscriptsubscript𝒬𝐹𝑆{\mathcal{Q}}_{F}^{S} be the set of (t:n)F𝒬F(t:n)_{F}\in{\mathcal{Q}}_{F} such that cvt𝔬vsubscript𝑐𝑣𝑡subscript𝔬𝑣c_{v}t\in\mathfrak{o}_{v} and cv2n𝔬v×subscriptsuperscript𝑐2𝑣𝑛superscriptsubscript𝔬𝑣c^{2}_{v}n\in\mathfrak{o}_{v}^{\times} for all vΣfinS𝑣subscriptΣfin𝑆v\in\Sigma_{\rm fin}-S with an idele c=(cv)(𝔸finS)×𝑐subscript𝑐𝑣superscriptsuperscriptsubscript𝔸fin𝑆c=(c_{v})\in({\mathbb{A}}_{\rm fin}^{S})^{\times} without S𝑆S-component.

Lemma 7.16.

Let Φ=vΦv\Phi=\otimes_{v}\Phi_{v} be our test function on G𝔸subscript𝐺𝔸G_{\mathbb{A}} (which depends l𝑙l, 𝔫𝔫{\mathfrak{n}} and S𝑆S). Let γGF𝛾subscript𝐺𝐹\gamma\in G_{F} be an element whose minimal polynomial is X2tX+nsuperscript𝑋2𝑡𝑋𝑛X^{2}-tX+n with (t,n)QF𝑡𝑛subscript𝑄𝐹(t,n)\in Q_{F}. If Φ(g1γg)0Φsuperscript𝑔1𝛾𝑔0\Phi(g^{-1}\gamma g)\not=0 with gG𝔸𝑔subscript𝐺𝔸g\in G_{\mathbb{A}}, then (t:n)F𝒬FS(t:n)_{F}\in{\mathcal{Q}}_{F}^{S}.

Proof.

Let vΣfinS𝑣subscriptΣfin𝑆v\in\Sigma_{\rm fin}-S. Then Φv=chZv𝕂0(𝔫𝔬v)subscriptΦ𝑣subscriptchsubscript𝑍𝑣subscript𝕂0𝔫subscript𝔬𝑣\Phi_{v}={\rm ch}_{Z_{v}{\mathbb{K}}_{0}({\mathfrak{n}}\mathfrak{o}_{v})}. Hence, from Φ(g1γg)0Φsuperscript𝑔1𝛾𝑔0\Phi(g^{-1}\gamma g)\not=0, we see that there exist gvGvsubscript𝑔𝑣subscript𝐺𝑣g_{v}\in G_{v}, cvFv×subscript𝑐𝑣superscriptsubscript𝐹𝑣c_{v}\in F_{v}^{\times} and kv𝕂0(𝔫𝔬v)subscript𝑘𝑣subscript𝕂0𝔫subscript𝔬𝑣k_{v}\in{\mathbb{K}}_{0}({\mathfrak{n}}\mathfrak{o}_{v}) such that gv1γgv=cvkvsuperscriptsubscript𝑔𝑣1𝛾subscript𝑔𝑣subscript𝑐𝑣subscript𝑘𝑣g_{v}^{-1}\gamma g_{v}=c_{v}k_{v}. Hence n=det(γ)=cv2det(kv)𝑛𝛾superscriptsubscript𝑐𝑣2subscript𝑘𝑣n=\det(\gamma)=c_{v}^{2}\,\det(k_{v}) and t=tr(γ)=cvtr(kv)𝑡tr𝛾subscript𝑐𝑣trsubscript𝑘𝑣t={\operatorname{tr}}(\gamma)=c_{v}\,{\operatorname{tr}}(k_{v}). Since detkv𝔬v×subscript𝑘𝑣superscriptsubscript𝔬𝑣\det k_{v}\in\mathfrak{o}_{v}^{\times} and tr(kv)𝔬vtrsubscript𝑘𝑣subscript𝔬𝑣{\operatorname{tr}}(k_{v})\in\mathfrak{o}_{v} and since n𝔬v×𝑛superscriptsubscript𝔬𝑣n\in\mathfrak{o}_{v}^{\times}, t𝔬v𝑡subscript𝔬𝑣t\in\mathfrak{o}_{v} for almost all v𝑣v’s, we have c=(cv)(𝔸finS)×𝑐subscript𝑐𝑣superscriptsuperscriptsubscript𝔸fin𝑆c=(c_{v})\in({\mathbb{A}}_{\rm fin}^{S})^{\times} and (t:n)F𝒬FS(t:n)_{F}\in{\mathcal{Q}}^{S}_{F}. ∎

Let hh be the class number of F𝐹F and 𝔞i(1ih)subscript𝔞𝑖1𝑖{\mathfrak{a}}_{i}\,(1\leqslant i\leqslant h) a complete set of representatives of the ideal class group of F𝐹F such that 𝔞j𝔬subscript𝔞𝑗𝔬{\mathfrak{a}}_{j}\subset\mathfrak{o} are prime ideals different from 𝔭v𝔬(vS)subscript𝔭𝑣𝔬𝑣𝑆{\mathfrak{p}}_{v}\cap{\mathfrak{o}}\,(v\in S). For ν=(νv)vSS𝜈subscriptsubscript𝜈𝑣𝑣𝑆superscript𝑆\nu=(\nu_{v})_{v\in S}\in{\mathbb{Z}}^{S}, set 𝔭Sν=vS(𝔭v𝔬)νvsuperscriptsubscript𝔭𝑆𝜈subscriptproduct𝑣𝑆superscriptsubscript𝔭𝑣𝔬subscript𝜈𝑣{\mathfrak{p}}_{S}^{\nu}=\prod_{v\in S}({\mathfrak{p}}_{v}\cap{\mathfrak{o}})^{\nu_{v}} and

Ei(S,ν)={nF×|n𝔬=𝔞i2𝔭Sν},(1ih).subscript𝐸𝑖𝑆𝜈conditional-set𝑛superscript𝐹𝑛𝔬superscriptsubscript𝔞𝑖2superscriptsubscript𝔭𝑆𝜈1𝑖E_{i}(S,\nu)=\left\{n\in F^{\times}|\,n\mathfrak{o}={\mathfrak{a}}_{i}^{2}{\mathfrak{p}}_{S}^{\nu}\,\right\},\quad(1\leqslant i\leqslant h).

Let I(S,ν)𝐼𝑆𝜈I(S,\nu) be the set of indices 1ih1𝑖1\leqslant i\leqslant h such that Ei(S,ν)subscript𝐸𝑖𝑆𝜈E_{i}(S,\nu)\not=\varnothing. For each iI(S,ν)𝑖𝐼𝑆𝜈i\in I(S,\nu), by fixing an element ni,νEi(S,ν)subscript𝑛𝑖𝜈subscript𝐸𝑖𝑆𝜈n_{i,\nu}\in E_{i}(S,\nu) once and for all, we obtain a bijection 𝔬×Ei(S,ν)superscript𝔬subscript𝐸𝑖𝑆𝜈\mathfrak{o}^{\times}\rightarrow E_{i}(S,\nu) by sending ε𝔬×𝜀superscript𝔬\varepsilon\in\mathfrak{o}^{\times} to εni,νEi(S,ν)𝜀subscript𝑛𝑖𝜈subscript𝐸𝑖𝑆𝜈\varepsilon n_{i,\nu}\in E_{i}(S,\nu). By Dirichlet’s unit theorem, the quotient group 𝔬×/(𝔬×)2superscript𝔬superscriptsuperscript𝔬2\mathfrak{o}^{\times}/(\mathfrak{o}^{\times})^{2} is finite.

In the following lemma, we set ordv(0)=+subscriptord𝑣0\operatorname{ord}_{v}(0)=+\infty for convention.

Lemma 7.17.

For any γ𝒬FS𝛾superscriptsubscript𝒬𝐹𝑆\gamma\in{\mathcal{Q}}_{F}^{S}, there exist ν0S𝜈superscriptsubscript0𝑆\nu\in{\mathbb{N}}_{0}^{S}, t𝔞i𝑡subscript𝔞𝑖t\in{\mathfrak{a}}_{i}, iI(S,ν)𝑖𝐼𝑆𝜈i\in I(S,\nu) and ε𝔬×/(𝔬×)2𝜀superscript𝔬superscriptsuperscript𝔬2\varepsilon\in\mathfrak{o}^{\times}/(\mathfrak{o}^{\times})^{2} such that γ=(t:εni,ν)F\gamma=(t:\varepsilon n_{i,\nu})_{F} and min(2ordv(t),νv){0,1}2subscriptord𝑣𝑡subscript𝜈𝑣01\min(2\operatorname{ord}_{v}(t),\nu_{v})\in\{0,1\} for all vS𝑣𝑆v\in S.

Proof.

To argue, we fix a representative (t,n)F2superscript𝑡superscript𝑛superscript𝐹2(t^{\prime},n^{\prime})\in F^{2} of γ𝛾\gamma. From definition, there exists a finite idele c=(cv)𝔸fin×𝑐subscript𝑐𝑣superscriptsubscript𝔸finc=(c_{v})\in{\mathbb{A}}_{\rm fin}^{\times} with cv=1(vS)subscript𝑐𝑣1𝑣𝑆c_{v}=1\,(v\in S) such that cvt𝔬vsubscript𝑐𝑣superscript𝑡subscript𝔬𝑣c_{v}t^{\prime}\in\mathfrak{o}_{v} and cv2n𝔬v×superscriptsubscript𝑐𝑣2superscript𝑛superscriptsubscript𝔬𝑣c_{v}^{2}n^{\prime}\in\mathfrak{o}_{v}^{\times} for all vΣfinS𝑣subscriptΣfin𝑆v\in\Sigma_{\rm fin}-S. Choose (av)vSvSFv×subscriptsubscript𝑎𝑣𝑣𝑆subscriptproduct𝑣𝑆superscriptsubscript𝐹𝑣(a_{v})_{v\in S}\in\prod_{v\in S}F_{v}^{\times} such that the ideal av2{(t)2𝔬v+n𝔬v}superscriptsubscript𝑎𝑣2superscriptsuperscript𝑡2subscript𝔬𝑣superscript𝑛subscript𝔬𝑣a_{v}^{2}\{(t^{\prime})^{2}\mathfrak{o}_{v}+n^{\prime}\mathfrak{o}_{v}\} is 𝔭vsubscript𝔭𝑣{\mathfrak{p}}_{v} or 𝔬vsubscript𝔬𝑣\mathfrak{o}_{v} for all vS𝑣𝑆v\in S. Let 𝔠𝔠{\mathfrak{c}} be a fractional ideal of F𝐹F defined by the idele ca𝑐𝑎ca, i.e., 𝔠=F(FvΣfinScv𝔬vvSav𝔬v)𝔠𝐹subscript𝐹subscriptproduct𝑣subscriptΣfin𝑆subscript𝑐𝑣subscript𝔬𝑣subscriptproduct𝑣𝑆subscript𝑎𝑣subscript𝔬𝑣{\mathfrak{c}}=F\cap(F_{\infty}\prod_{v\in\Sigma_{\rm fin}-S}c_{v}\mathfrak{o}_{v}\prod_{v\in S}a_{v}\mathfrak{o}_{v}). There exists a unique i{1,,h}𝑖1i\in\{1,\dots,h\} and an element bF×𝑏superscript𝐹b\in F^{\times} such that 𝔠=(b)𝔞i1𝔠𝑏superscriptsubscript𝔞𝑖1{\mathfrak{c}}=(b){\mathfrak{a}}_{i}^{-1}. We set t′′=btsuperscript𝑡′′𝑏superscript𝑡t^{\prime\prime}=bt^{\prime} and n′′=b2nsuperscript𝑛′′superscript𝑏2superscript𝑛n^{\prime\prime}=b^{2}n^{\prime}. Then γ=(t:n)F=(t′′:n′′)F\gamma=(t^{\prime}:n^{\prime})_{F}=(t^{\prime\prime}:n^{\prime\prime})_{F}, t′′𝔞isuperscript𝑡′′subscript𝔞𝑖t^{\prime\prime}\in{\mathfrak{a}}_{i} and n′′Ei(S,ν)superscript𝑛′′subscript𝐸𝑖𝑆𝜈n^{\prime\prime}\in E_{i}(S,\nu) with some ν0S𝜈superscriptsubscript0𝑆\nu\in{\mathbb{N}}_{0}^{S} such that min(2ordv(t′′),νv){0,1}2subscriptord𝑣superscript𝑡′′subscript𝜈𝑣01\min(2\operatorname{ord}_{v}(t^{\prime\prime}),\nu_{v})\in\{0,1\} for all vS𝑣𝑆v\in S. We further write n′′=εu2ni,νsuperscript𝑛′′𝜀superscript𝑢2subscript𝑛𝑖𝜈n^{\prime\prime}=\varepsilon u^{2}n_{i,\nu} with ε𝔬×/(𝔬×)2𝜀superscript𝔬superscriptsuperscript𝔬2\varepsilon\in\mathfrak{o}^{\times}/(\mathfrak{o}^{\times})^{2}, u𝔬×𝑢superscript𝔬u\in\mathfrak{o}^{\times}, and set t=u1t′′𝑡superscript𝑢1superscript𝑡′′t=u^{-1}t^{\prime\prime}. Then γ=(t′′:n′′)F=(t:εni,ν)F\gamma=(t^{\prime\prime}:n^{\prime\prime})_{F}=(t:\varepsilon n_{i,\nu})_{F} and t𝔞i𝑡subscript𝔞𝑖t\in{\mathfrak{a}}_{i} with min(2ordv(t),νv)=0,12subscriptord𝑣𝑡subscript𝜈𝑣01\min(2\operatorname{ord}_{v}(t),\nu_{v})=0,1 for all vS𝑣𝑆v\in S as desired. ∎

For (t:n)F𝒬FIrr(t:n)_{F}\in{\mathcal{Q}}_{F}^{\rm{Irr}} and a place vΣfin𝑣subscriptΣfinv\in\Sigma_{{\rm fin}}, by (t24n)(v)=4Δv0mv2superscriptsuperscript𝑡24𝑛𝑣4superscriptsubscriptΔ𝑣0superscriptsubscript𝑚𝑣2(t^{2}-4n)^{(v)}=4\Delta_{v}^{0}m_{v}^{2} as before, we set 𝔫v(t,n)=n(v)mv2subscript𝔫𝑣𝑡𝑛superscript𝑛𝑣superscriptsubscript𝑚𝑣2{\mathfrak{n}}_{v}(t,n)=\frac{n^{(v)}}{m_{v}^{2}}, which is an element of Fvsubscript𝐹𝑣F_{v} determined only up to proportionality constants from 𝔬v×superscriptsubscript𝔬𝑣\mathfrak{o}_{v}^{\times}. We remark that 𝔈v(z)(γ^v)=0superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣0{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=0 for vΣfinS𝑣subscriptΣfin𝑆v\in\Sigma_{\rm fin}-S unless |𝔫v(t,n)|v1subscriptsubscript𝔫𝑣𝑡𝑛𝑣1|{\mathfrak{n}}_{v}(t,n)|_{v}\geqslant 1 from Lemma 7.11 and Theorem 7.9. For the arguments below to work, we need to make the following assumption (7.11) so that any representative (t,εni,ν)𝑡𝜀subscript𝑛𝑖𝜈(t,\varepsilon n_{i,\nu}) as in Lemma 7.17 satisfies the condition (a) in Lemma 7.15:

(7.11) The ideal 𝔫𝔫{\mathfrak{n}} is relatively prime to all of the ideals 𝔞j(1jh)subscript𝔞𝑗1𝑗{\mathfrak{a}}_{j}\,(1\leqslant j\leqslant h).

We fix prime ideals 𝔞i(1ih)subscript𝔞𝑖1𝑖{\mathfrak{a}}_{i}(1\leqslant i\leqslant h) satisfying (7.11) for 𝔫𝔫{\mathfrak{n}} invoking Chebotarev’s density theorem. From Lemmas 7.16 and 7.17, we have a majorant of the series (7.10) by replacing the summation range with the set of all those pairs (t,n)𝑡𝑛(t,n) with t𝔞i𝑡subscript𝔞𝑖t\in{\mathfrak{a}}_{i}, n=εni,ν𝑛𝜀subscript𝑛𝑖𝜈n=\varepsilon n_{i,\nu} with ε𝔬×/(𝔬×)2𝜀superscript𝔬superscriptsuperscript𝔬2\varepsilon\in\mathfrak{o}^{\times}/(\mathfrak{o}^{\times})^{2}, 1ih1𝑖1\leqslant i\leqslant h and ν0S𝜈superscriptsubscript0𝑆\nu\in{\mathbb{N}}_{0}^{S}. Let ϵ>0italic-ϵ0\epsilon>0 be a small number. In the following all estimations are uniform in z𝑧z\in{\mathbb{C}} such that Re(z)[2l¯+3+6ϵ,l¯1/2ϵ]Re𝑧2¯𝑙36italic-ϵ¯𝑙12italic-ϵ\operatorname{Re}(z)\in[-2\underline{l}+3+6\epsilon,\underline{l}-1/2-\epsilon]. By the evaluations of local orbital integrals 𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) obtained in this section and by Lemmas 7.8 and 7.15, we see that (7.10) is majorized by the sum of Ξ(z,𝐬)(𝔠,𝔫1,i,n)superscriptΞ𝑧𝐬𝔠subscript𝔫1𝑖𝑛\Xi^{(z,{\mathbf{s}})}({\mathfrak{c}},{\mathfrak{n}}_{1},i,n) over all tuples (𝔠,𝔫1,i,n)𝔠subscript𝔫1𝑖𝑛({\mathfrak{c}},{\mathfrak{n}}_{1},i,n) of an integral ideal 𝔫1subscript𝔫1{\mathfrak{n}}_{1} dividing 𝔫𝔫{\mathfrak{n}}, an integral ideal 𝔠𝔠{\mathfrak{c}} dividing 𝔫1subscript𝔫1{\mathfrak{n}}_{1}, 1ih1𝑖1\leqslant i\leqslant h, and an element nF×𝑛superscript𝐹n\in F^{\times} which is of the form n=εni,ν𝑛𝜀subscript𝑛𝑖𝜈n=\varepsilon n_{i,\nu} with ε𝔬×/(𝔬×)2𝜀superscript𝔬superscriptsuperscript𝔬2\varepsilon\in\mathfrak{o}^{\times}/(\mathfrak{o}^{\times})^{2}, iI(S,ν)𝑖𝐼𝑆𝜈i\in I(S,\nu) and ν0S𝜈superscriptsubscript0𝑆\nu\in{\mathbb{N}}_{0}^{S}, where Ξ(z,𝐬)(𝔠,𝔫1,i,n)superscriptΞ𝑧𝐬𝔠subscript𝔫1𝑖𝑛\Xi^{(z,{\mathbf{s}})}({\mathfrak{c}},{\mathfrak{n}}_{1},i,n) is defined to be

tX(𝔠,𝔫1,i,n)N(𝔇t24n)1+ϵ4+ϱ(z)4vΣfin(SS(𝔫))|𝒪v,0Δv0,(z)(𝔫v(t,n))|vS(𝔫)|𝒪v,1Δv0,(z)(𝔫v(t,n))|\displaystyle\sum_{t\in X({\mathfrak{c}},{\mathfrak{n}}_{1},i,n)}{\operatorname{N}}({\mathfrak{D}}_{t^{2}-4n})^{\frac{1+\epsilon}{4}+\frac{\varrho(z)}{4}}\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}|{\mathcal{O}}_{v,0}^{\Delta_{v}^{0},(z)}({\mathfrak{n}}_{v}(t,n))|\prod_{v\in S({\mathfrak{n}})}|{\mathcal{O}}_{v,1}^{\Delta_{v}^{0},(z)}({\mathfrak{n}}_{v}(t,n))|
×vS|𝒮vΔv0,(z)(sv;𝔫v(t,n))|vΣ(t24n)(v)>0|𝒪v+,(z)(t(v)|t24n|v1/2)|vΣ(t24n)(v)<0|𝒪v,(z)(t(v)|t24n|v1/2)|\displaystyle\times\prod_{v\in S}|{\mathcal{S}}_{v}^{\Delta_{v}^{0},(z)}(s_{v};{\mathfrak{n}}_{v}(t,n))|\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}>0\end{subarray}}\left|{\mathcal{O}}_{v}^{+,(z)}\left(\tfrac{t^{(v)}}{|t^{2}-4n|_{v}^{1/2}}\right)\right|\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}<0\end{subarray}}\left|{\mathcal{O}}_{v}^{-,(z)}\left(\tfrac{t^{(v)}}{|t^{2}-4n|_{v}^{1/2}}\right)\right|

with X(𝔠,𝔫1,i,n)𝑋𝔠subscript𝔫1𝑖𝑛X({\mathfrak{c}},{\mathfrak{n}}_{1},i,n) being the set of all t𝔞i𝔠R(n)𝑡subscript𝔞𝑖𝔠𝑅𝑛t\in{\mathfrak{a}}_{i}{\mathfrak{c}}-R(n) such that t(v)𝔬v×(vS(𝔫𝔫11))superscript𝑡𝑣superscriptsubscript𝔬𝑣for-all𝑣𝑆𝔫superscriptsubscript𝔫11t^{(v)}\in\mathfrak{o}_{v}^{\times}\,(\forall v\in S({\mathfrak{n}}{\mathfrak{n}}_{1}^{-1})), n(v)𝔬v×(vS(𝔫𝔫11))superscript𝑛𝑣superscriptsubscript𝔬𝑣for-all𝑣𝑆𝔫superscriptsubscript𝔫11n^{(v)}\in\mathfrak{o}_{v}^{\times}\,(\forall v\in S({\mathfrak{n}}{\mathfrak{n}}_{1}^{-1})), and such that t24n(𝔠1𝔫1)2superscript𝑡24𝑛superscriptsuperscript𝔠1subscript𝔫12t^{2}-4n\in({\mathfrak{c}}^{-1}{\mathfrak{n}}_{1})^{2}, (t24n)(v)(𝔬v×)2(vS(𝔠𝔫𝔫11))superscriptsuperscript𝑡24𝑛𝑣superscriptsuperscriptsubscript𝔬𝑣2for-all𝑣𝑆𝔠𝔫superscriptsubscript𝔫11(t^{2}-4n)^{(v)}\in(\mathfrak{o}_{v}^{\times})^{2}\,(\forall v\in S({\mathfrak{c}}{\mathfrak{n}}{\mathfrak{n}}_{1}^{-1})). Here 𝐬=(sv)vS𝔛S𝐬subscriptsubscript𝑠𝑣𝑣𝑆subscript𝔛𝑆{\mathbf{s}}=(s_{v})_{v\in S}\in{\mathfrak{X}}_{S} and we remark that Δv0superscriptsubscriptΔ𝑣0\Delta_{v}^{0} depends on t,n𝑡𝑛t,n through the relation (t24n)(v)=4Δv0mv2superscriptsuperscript𝑡24𝑛𝑣4superscriptsubscriptΔ𝑣0superscriptsubscript𝑚𝑣2(t^{2}-4n)^{(v)}=4\Delta_{v}^{0}m_{v}^{2}. For n=εni,ν𝑛𝜀subscript𝑛𝑖𝜈n=\varepsilon n_{i,\nu} as above, the subset R(n)𝔬𝑅𝑛𝔬R(n)\subset\mathfrak{o} is defined as follows: R(n)={0}𝑅𝑛0R(n)=\{0\} if n(v)mod𝔭vmodulosuperscript𝑛𝑣subscript𝔭𝑣n^{(v)}\mod{\mathfrak{p}}_{v} is not a square residue for some vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}), and R(n)=𝑅𝑛R(n)=\varnothing otherwise. This comes from Lemma 7.15.

From Lemmas 7.10 and 7.11, if vΣfinΣdyadic𝑣subscriptΣfinsubscriptΣdyadicv\in\Sigma_{\rm fin}-\Sigma_{\rm dyadic}, t𝔬v×𝑡superscriptsubscript𝔬𝑣t\in\mathfrak{o}_{v}^{\times}, 4n𝔬v×,4nt21𝔬v×formulae-sequence4𝑛superscriptsubscript𝔬𝑣4𝑛superscript𝑡21superscriptsubscript𝔬𝑣4n\in\mathfrak{o}_{v}^{\times},\,\tfrac{4n}{t^{2}}-1\in\mathfrak{o}_{v}^{\times}, then |𝔱v(t,n)|v=1subscriptsubscript𝔱𝑣𝑡𝑛𝑣1|{\mathfrak{t}}_{v}(t,n)|_{v}=1 and |𝔫v(t,n)|v=1subscriptsubscript𝔫𝑣𝑡𝑛𝑣1|{\mathfrak{n}}_{v}(t,n)|_{v}=1. In a way similar to Corollary 6.6, from Lemma 7.14, we have

vΣfin(SS(𝔫))|𝒪v,0Δv0,(z)(𝔫v(t,n))|CvΣfin(SS(𝔫))|𝔫v(t,n)|v|Re(z)|+14+ϵ2subscriptproduct𝑣subscriptΣfin𝑆𝑆𝔫superscriptsubscript𝒪𝑣0superscriptsubscriptΔ𝑣0𝑧subscript𝔫𝑣𝑡𝑛𝐶subscriptproduct𝑣subscriptΣfin𝑆𝑆𝔫superscriptsubscriptsubscript𝔫𝑣𝑡𝑛𝑣Re𝑧14italic-ϵ2\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}|{\mathcal{O}}_{v,0}^{\Delta_{v}^{0},(z)}({\mathfrak{n}}_{v}(t,n))|\leqslant C\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}|{\mathfrak{n}}_{v}(t,n)|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}

with a constant C>0𝐶0C>0, uniformly in z𝑧z\in{\mathbb{C}} and (t,n)𝑡𝑛(t,n) as above.

Suppose R(n)={0}𝑅𝑛0R(n)=\{0\} for a while. Applying Lemma 7.14, we see that Ξ(z,𝐬)(𝔠,𝔫1,i,n)superscriptΞ𝑧𝐬𝔠subscript𝔫1𝑖𝑛\Xi^{(z,{\mathbf{s}})}({\mathfrak{c}},{\mathfrak{n}}_{1},i,n) is majorized by the product of

tX(𝔠,𝔫1,i,n)N(𝔇t24n)1+ϵ4+ϱ(z)4×vΣfin(SS(𝔫))|𝔫v(t,n)|v|Re(z)|+14+ϵ2vS(𝔫)|𝔫v(t,n)|v|Re(z)|+14+ϵ2\displaystyle\sum_{t\in X({\mathfrak{c}},{\mathfrak{n}}_{1},i,n)}{\operatorname{N}}({\mathfrak{D}}_{t^{2}-4n})^{\frac{1+\epsilon}{4}+\frac{\varrho(z)}{4}}\times\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}|{\mathfrak{n}}_{v}(t,n)|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}\prod_{v\in S({\mathfrak{n}})}|{\mathfrak{n}}_{v}(t,n)|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}
×vS{1+max(0,ordv(𝔫v(t,n)))}max(1,|𝔫v(t,n)|v1)Re(sv)+|Re(z)|4+ϵ2|𝔫v(t,n)|v|Re(z)|+14+ϵ2\displaystyle\times\prod_{v\in S}\{1+\max(0,-\operatorname{ord}_{v}({\mathfrak{n}}_{v}(t,n)))\}\max(1,|{\mathfrak{n}}_{v}(t,n)|_{v}^{-1})^{\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}|{\mathfrak{n}}_{v}(t,n)|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}
×vΣ(t24n)(v)>0δ(|t|v2>|t24n|v)(|t(v)(t(v))24n(v)1|v+)lv/2(|t|v|t24n|v1/2)|Re(z)|+12\displaystyle\times\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}>0\end{subarray}}\delta(|t|_{v}^{2}>|t^{2}-4n|_{v})\,\left(\left|\frac{t^{(v)}}{\sqrt{(t^{(v)})^{2}-4n^{(v)}}}-1\right|_{v}^{+}\right)^{l_{v}/2}\left(\frac{|t|_{v}}{|t^{2}-4n|_{v}^{1/2}}\right)^{\frac{|\operatorname{Re}(z)|+1}{2}}
×vΣ(t24n)(v)<0(1+|t|v|t24n|v1/2)12lv4×vS(𝔫𝔫11)2qv+1vS(𝔫1)4(1+qv1/2)1+qv,\displaystyle\times\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}<0\end{subarray}}\left(1+\frac{|t|_{v}}{|t^{2}-4n|_{v}^{1/2}}\right)^{\frac{1-2l_{v}}{4}}\times\prod_{v\in S({\mathfrak{n}}{\mathfrak{n}}_{1}^{-1})}\frac{2}{q_{v}+1}\prod_{v\in S({\mathfrak{n}}_{1})}\frac{4(1+q_{v}^{1/2})}{1+q_{v}},

where we set |x|v+=min(1,|x|v)superscriptsubscript𝑥𝑣1subscript𝑥𝑣|x|_{v}^{+}=\min(1,|x|_{v}) for xFv×𝑥superscriptsubscript𝐹𝑣x\in F_{v}^{\times}. Since N(𝔇t24n)=vΣfin|Δv0|v1=vΣF|mv|v2Nsubscript𝔇superscript𝑡24𝑛subscriptproduct𝑣subscriptΣfinsuperscriptsubscriptsuperscriptsubscriptΔ𝑣0𝑣1subscriptproduct𝑣subscriptΣ𝐹superscriptsubscriptsubscript𝑚𝑣𝑣2{\operatorname{N}}({\mathfrak{D}}_{t^{2}-4n})=\prod_{v\in\Sigma_{\rm fin}}|\Delta_{v}^{0}|_{v}^{-1}=\prod_{v\in\Sigma_{F}}|m_{v}|_{v}^{2} (see the last part of the proof of Proposition 7.7), it is easy to confirm the identity

N(𝔇t24n)|Re(z)|+14+ϵ2vΣfin|𝔫v(t,n)|v|Re(z)|+14+ϵ2=vΣ|n|v|Re(z)|+14ϵ2|41(t24n)|v|Re(z)|+14+ϵ2\displaystyle{\operatorname{N}}({\mathfrak{D}}_{t^{2}-4n})^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}\,\prod_{v\in\Sigma_{\rm fin}}|{\mathfrak{n}}_{v}(t,n)|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}=\prod_{v\in\Sigma_{\infty}}|n|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{4}-\frac{\epsilon}{2}}|4^{-1}(t^{2}-4n)|_{v}^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}

by the product formula and by the relation |mv|v=|41(t24n)|v1/2subscriptsubscript𝑚𝑣𝑣superscriptsubscriptsuperscript41superscript𝑡24𝑛𝑣12|m_{v}|_{v}=|4^{-1}(t^{2}-4n)|_{v}^{1/2} for vΣ𝑣subscriptΣv\in\Sigma_{\infty}. Due to this, the above quantity is further majorized by

vS(𝔫𝔫11)2qv+1vS(𝔫1)4(1+qv1/2)1+qvtX(𝔠,𝔫1,i,n)N(𝔇t24n)ϱ(z)|Re(z)|4ϵ4\displaystyle\prod_{v\in S({\mathfrak{n}}{\mathfrak{n}}_{1}^{-1})}\frac{2}{q_{v}+1}\,\prod_{v\in S({\mathfrak{n}}_{1})}\frac{4(1+q_{v}^{1/2})}{1+q_{v}}\sum_{t\in X({\mathfrak{c}},{\mathfrak{n}}_{1},i,n)}{\operatorname{N}}({\mathfrak{D}}_{t^{2}-4n})^{\frac{\varrho(z)-|\operatorname{Re}(z)|}{4}-\frac{\epsilon}{4}}
×vS{1+max(0,ordv(𝔫v(t,n)))}max(1,|𝔫v(t,n)|v1)Re(sv)+|Re(z)|4+ϵ2\displaystyle\times\prod_{v\in S}\{1+\max(0,-\operatorname{ord}_{v}({\mathfrak{n}}_{v}(t,n)))\}\max(1,|{\mathfrak{n}}_{v}(t,n)|_{v}^{-1})^{\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}
×vΣ(t24n)(v)>0δ(n(v)>0)(|t(v)(t(v))24n(v)1|v+)lv/2\displaystyle\times\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}>0\end{subarray}}\delta(n^{(v)}>0)\,\left(\left|\frac{t^{(v)}}{\sqrt{(t^{(v)})^{2}-4n^{(v)}}}-1\right|_{v}^{+}\right)^{l_{v}/2}
×vΣ(|t|v2|n|v)|Re(z)|+14+ϵ2|t24n|vϵ2vΣ(t24n)(v)<0(1+|t|v|t24n|v1/2)12lv4(|t|v|t24n|v1/2)1+|Re(z)|2ϵ.\displaystyle\times\prod_{v\in\Sigma_{\infty}}\left(\frac{|t|_{v}^{2}}{|n|_{v}}\right)^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}|t^{2}-4n|_{v}^{\frac{\epsilon}{2}}\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}<0\end{subarray}}\left(1+\frac{|t|_{v}}{|t^{2}-4n|_{v}^{1/2}}\right)^{\frac{1-2l_{v}}{4}}\left(\frac{|t|_{v}}{|t^{2}-4n|_{v}^{1/2}}\right)^{-\frac{1+|\operatorname{Re}(z)|}{2}-\epsilon}.

Let us examine the S𝑆S-factor. If νv=0subscript𝜈𝑣0\nu_{v}=0, then (4n)(v)𝔬v×superscript4𝑛𝑣superscriptsubscript𝔬𝑣(4n)^{(v)}\in\mathfrak{o}_{v}^{\times} and mv𝔬vsubscript𝑚𝑣subscript𝔬𝑣m_{v}\in{\mathfrak{o}}_{v}. Hence |𝔫v(t,n)|v=|mv2|v11subscriptsubscript𝔫𝑣𝑡𝑛𝑣superscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣11|{\mathfrak{n}}_{v}(t,n)|_{v}=|m_{v}^{2}|_{v}^{-1}\geqslant 1. If νv>0subscript𝜈𝑣0\nu_{v}>0, then (4n)(v)𝔭vsuperscript4𝑛𝑣subscript𝔭𝑣(4n)^{(v)}\in{\mathfrak{p}}_{v}. When t(v)𝔬v×superscript𝑡𝑣superscriptsubscript𝔬𝑣t^{(v)}\in\mathfrak{o}_{v}^{\times}, we have |t|v2>|4n|vsuperscriptsubscript𝑡𝑣2subscript4𝑛𝑣|t|_{v}^{2}>|4n|_{v} and t24n𝔬v×superscript𝑡24𝑛superscriptsubscript𝔬𝑣t^{2}-4n\in\mathfrak{o}_{v}^{\times} is a square residue modulo 𝔭vsubscript𝔭𝑣{\mathfrak{p}}_{v}. Since v𝑣v is non-dyadic, this implies (t24n)(v)superscriptsuperscript𝑡24𝑛𝑣(t^{2}-4n)^{(v)} is a square in 𝔬v×superscriptsubscript𝔬𝑣\mathfrak{o}_{v}^{\times}, or equivalently Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1 and 2mv𝔬v×2subscript𝑚𝑣superscriptsubscript𝔬𝑣2m_{v}\in\mathfrak{o}_{v}^{\times}. When t(v)𝔭vsuperscript𝑡𝑣subscript𝔭𝑣t^{(v)}\in{\mathfrak{p}}_{v}, we have νv=1subscript𝜈𝑣1\nu_{v}=1 by min(2ord(t),νv){0,1}2ord𝑡subscript𝜈𝑣01\min(2\operatorname{ord}(t),\nu_{v})\in\{0,1\}. Thus ordv((2mv)2Δv0)=ordv(t24n)=1subscriptord𝑣superscript2subscript𝑚𝑣2superscriptsubscriptΔ𝑣0subscriptord𝑣superscript𝑡24𝑛1\operatorname{ord}_{v}((2m_{v})^{2}\Delta_{v}^{0})=\operatorname{ord}_{v}(t^{2}-4n)=1, and hence 2mv𝔬×2subscript𝑚𝑣superscript𝔬2m_{v}\in{\mathfrak{o}}^{\times}. Thus we have |𝔫v(t,n)|v=|n|v|mv2|v1=|n|v=qvνvsubscriptsubscript𝔫𝑣𝑡𝑛𝑣subscript𝑛𝑣superscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣1subscript𝑛𝑣superscriptsubscript𝑞𝑣subscript𝜈𝑣|{\mathfrak{n}}_{v}(t,n)|_{v}=|n|_{v}|m_{v}^{2}|_{v}^{-1}=|n|_{v}=q_{v}^{-\nu_{v}} if νv>0subscript𝜈𝑣0\nu_{v}>0. Therefore, ,

vS{1+max(0,ordv(𝔫v(t,n)))}max(1,|𝔫v(t,n)|v1)Re(sv)+|Re(z)|4+ϵ2\displaystyle\prod_{v\in S}\{1+\max(0,-\operatorname{ord}_{v}({\mathfrak{n}}_{v}(t,n)))\}\max(1,|{\mathfrak{n}}_{v}(t,n)|_{v}^{-1})^{\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}
\displaystyle\leqslant vS(1+ordv(mv2))δ(νv=0)qvνv(Re(sv)+|Re(z)|4+ϵ2)vS|mv2|vϵ4qvνv(Re(sv)+|Re(z)|4+ϵ2).much-less-thansubscriptproduct𝑣𝑆superscript1subscriptord𝑣superscriptsubscript𝑚𝑣2𝛿subscript𝜈𝑣0superscriptsubscript𝑞𝑣subscript𝜈𝑣Resubscript𝑠𝑣Re𝑧4italic-ϵ2subscriptproduct𝑣𝑆superscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣italic-ϵ4superscriptsubscript𝑞𝑣subscript𝜈𝑣Resubscript𝑠𝑣Re𝑧4italic-ϵ2\displaystyle\prod_{v\in S}(1+\operatorname{ord}_{v}(m_{v}^{2}))^{\delta(\nu_{v}=0)}q_{v}^{\nu_{v}(\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2})}\ll\prod_{v\in S}|m_{v}^{2}|_{v}^{-\frac{\epsilon}{4}}q_{v}^{\nu_{v}(\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2})}.

Here, by noting 1=|Δ/4|𝔸=vΣF|mv2|vvΣfin|Δv0|v1subscriptΔ4𝔸subscriptproduct𝑣subscriptΣ𝐹subscriptsuperscriptsubscript𝑚𝑣2𝑣subscriptproduct𝑣subscriptΣfinsubscriptsuperscriptsubscriptΔ𝑣0𝑣1=|\Delta/4|_{\mathbb{A}}=\prod_{v\in\Sigma_{F}}|m_{v}^{2}|_{v}\prod_{v\in\Sigma_{\rm fin}}|\Delta_{v}^{0}|_{v}, the factor vS(1+ordv(mv2))subscriptproduct𝑣𝑆1subscriptord𝑣superscriptsubscript𝑚𝑣2\prod_{v\in S}(1+\operatorname{ord}_{v}(m_{v}^{2})) is bounded by

vS|mv2|vϵ4=subscriptproduct𝑣𝑆superscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣italic-ϵ4absent\displaystyle\prod_{v\in S}|m_{v}^{2}|_{v}^{-\frac{\epsilon}{4}}= vΣF|mv2|vϵ4×vΣ|mv2|vϵ4×vΣfinS|mv2|vϵ4subscriptproduct𝑣subscriptΣ𝐹superscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣italic-ϵ4subscriptproduct𝑣subscriptΣsuperscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣italic-ϵ4subscriptproduct𝑣subscriptΣfin𝑆superscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣italic-ϵ4\displaystyle\prod_{v\in\Sigma_{F}}|m_{v}^{2}|_{v}^{-\frac{\epsilon}{4}}\times\prod_{v\in\Sigma_{\infty}}|m_{v}^{2}|_{v}^{\frac{\epsilon}{4}}\times\prod_{v\in\Sigma_{\rm fin}-S}|m_{v}^{2}|_{v}^{\frac{\epsilon}{4}}
\displaystyle\leqslant vΣfin|Δv0|vϵ4×vΣ|mv2|vϵ4×vΣfinS|41(Δv0)1|vϵ4subscriptproduct𝑣subscriptΣfinsuperscriptsubscriptsuperscriptsubscriptΔ𝑣0𝑣italic-ϵ4subscriptproduct𝑣subscriptΣsuperscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣italic-ϵ4subscriptproduct𝑣subscriptΣfin𝑆superscriptsubscriptsuperscript41superscriptsuperscriptsubscriptΔ𝑣01𝑣italic-ϵ4\displaystyle\prod_{v\in\Sigma_{{\rm fin}}}|\Delta_{v}^{0}|_{v}^{\frac{\epsilon}{4}}\times\prod_{v\in\Sigma_{\infty}}|m_{v}^{2}|_{v}^{\frac{\epsilon}{4}}\times\prod_{v\in\Sigma_{\rm fin}-S}|4^{-1}(\Delta_{v}^{0})^{-1}|_{v}^{\frac{\epsilon}{4}}
=\displaystyle= vS|Δv0|vϵ4×vΣ|mv2|vϵ4×vΣfinS|41|vϵ4subscriptproduct𝑣𝑆superscriptsubscriptsuperscriptsubscriptΔ𝑣0𝑣italic-ϵ4subscriptproduct𝑣subscriptΣsuperscriptsubscriptsuperscriptsubscript𝑚𝑣2𝑣italic-ϵ4subscriptproduct𝑣subscriptΣfin𝑆superscriptsubscriptsuperscript41𝑣italic-ϵ4\displaystyle\prod_{v\in S}|\Delta_{v}^{0}|_{v}^{\frac{\epsilon}{4}}\times\prod_{v\in\Sigma_{\infty}}|m_{v}^{2}|_{v}^{\frac{\epsilon}{4}}\times\prod_{v\in\Sigma_{\rm fin}-S}|4^{-1}|_{v}^{\frac{\epsilon}{4}}
Ssubscriptmuch-less-than𝑆\displaystyle\ll_{S} vΣ|t24n|vϵ4.subscriptproduct𝑣subscriptΣsuperscriptsubscriptsuperscript𝑡24𝑛𝑣italic-ϵ4\displaystyle\prod_{v\in\Sigma_{\infty}}|t^{2}-4n|_{v}^{\frac{\epsilon}{4}}.

By this and by the majorization N(𝔇t24n)vΣ|t24n|vmuch-less-thanNsubscript𝔇superscript𝑡24𝑛subscriptproduct𝑣subscriptΣsubscriptsuperscript𝑡24𝑛𝑣{\operatorname{N}}({\mathfrak{D}}_{t^{2}-4n})\ll\prod_{v\in\Sigma_{\infty}}|t^{2}-4n|_{v}, we have that Ξ(z,𝐬)(𝔠,𝔫1,i,n)superscriptΞ𝑧𝐬𝔠subscript𝔫1𝑖𝑛\Xi^{(z,{\mathbf{s}})}({\mathfrak{c}},{\mathfrak{n}}_{1},i,n) is majorized by the product of

vS(𝔫𝔫11)21+qvvS(𝔫1)4(1+qv1/2)1+qvvSqvνv(Re(sv)+|Re(z)|4+ϵ2)subscriptproduct𝑣𝑆𝔫superscriptsubscript𝔫1121subscript𝑞𝑣subscriptproduct𝑣𝑆subscript𝔫141superscriptsubscript𝑞𝑣121subscript𝑞𝑣subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣subscript𝜈𝑣Resubscript𝑠𝑣Re𝑧4italic-ϵ2\displaystyle\prod_{v\in S({\mathfrak{n}}{\mathfrak{n}}_{1}^{-1})}\frac{2}{1+q_{v}}\,\prod_{v\in S({\mathfrak{n}}_{1})}\frac{4(1+q_{v}^{1/2})}{1+q_{v}}\,\prod_{v\in S}q_{v}^{\nu_{v}(\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2})}

and

(7.12) t𝔞i𝔠{0}t24n(𝔫1𝔠1)2vΣ(t24n)(v)>0δ(n(v)>0)(|t(v)(t(v))24n(v)1|v+)lv/2subscript𝑡subscript𝔞𝑖𝔠0superscript𝑡24𝑛superscriptsubscript𝔫1superscript𝔠12subscriptproduct𝑣subscriptΣsuperscriptsuperscript𝑡24𝑛𝑣0𝛿superscript𝑛𝑣0superscriptsuperscriptsubscriptsuperscript𝑡𝑣superscriptsuperscript𝑡𝑣24superscript𝑛𝑣1𝑣subscript𝑙𝑣2\displaystyle\sum_{\begin{subarray}{c}t\in{\mathfrak{a}}_{i}{\mathfrak{c}}-\{0\}\\ t^{2}-4n\in({\mathfrak{n}}_{1}{\mathfrak{c}}^{-1})^{2}\end{subarray}}\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}>0\end{subarray}}\delta(n^{(v)}>0)\,\left(\left|\tfrac{t^{(v)}}{\sqrt{(t^{(v)})^{2}-4n^{(v)}}}-1\right|_{v}^{+}\right)^{l_{v}/2}
vΣ|t24n|vϱ(z)|Re(z)|4+ϵ2(|t|v2|n|v)|Re(z)|+14+ϵ2×vΣ(t24n)(v)<0(1+|t|v|t24n|v1/2)12lv4(|t|v|t24n|v1/2)1+|Re(z)|2ϵ.subscriptproduct𝑣subscriptΣsuperscriptsubscriptsuperscript𝑡24𝑛𝑣italic-ϱ𝑧Re𝑧4italic-ϵ2superscriptsuperscriptsubscript𝑡𝑣2subscript𝑛𝑣Re𝑧14italic-ϵ2subscriptproduct𝑣subscriptΣsuperscriptsuperscript𝑡24𝑛𝑣0superscript1subscript𝑡𝑣superscriptsubscriptsuperscript𝑡24𝑛𝑣1212subscript𝑙𝑣4superscriptsubscript𝑡𝑣superscriptsubscriptsuperscript𝑡24𝑛𝑣121Re𝑧2italic-ϵ\displaystyle\prod_{v\in\Sigma_{\infty}}|t^{2}-4n|_{v}^{\frac{\varrho(z)-|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}\,\left(\tfrac{|t|_{v}^{2}}{|n|_{v}}\right)^{\frac{|\operatorname{Re}(z)|+1}{4}+\frac{\epsilon}{2}}\times\prod_{\begin{subarray}{c}v\in\Sigma_{\infty}\\ (t^{2}-4n)^{(v)}<0\end{subarray}}\left(1+\tfrac{|t|_{v}}{|t^{2}-4n|_{v}^{1/2}}\right)^{\frac{1-2l_{v}}{4}}\left(\tfrac{|t|_{v}}{|t^{2}-4n|_{v}^{1/2}}\right)^{-\frac{1+|\operatorname{Re}(z)|}{2}-\epsilon}.

By the embedding ι:t(t(v))vΣ:subscript𝜄maps-to𝑡subscriptsuperscript𝑡𝑣𝑣subscriptΣ\iota_{\infty}:t\mapsto(t^{(v)})_{v\in\Sigma_{\infty}}, any fractional ideal of F𝐹F is viewed as a {\mathbb{Z}}-lattice of [F:]delimited-[]:𝐹[F:{\mathbb{Q}}]-dimensional real vector space F=vΣFvsubscript𝐹subscriptproduct𝑣subscriptΣsubscript𝐹𝑣F_{\infty}=\prod_{v\in\Sigma_{\infty}}F_{v}. Let us examine the condition t24n(𝔫1𝔠1)2superscript𝑡24𝑛superscriptsubscript𝔫1superscript𝔠12t^{2}-4n\in({\mathfrak{n}}_{1}{\mathfrak{c}}^{-1})^{2}. Set dF=[F:]d_{F}=[F:{\mathbb{Q}}]. We use the Euclidean norm ξ=(vΣxv2)1/2norm𝜉superscriptsubscript𝑣subscriptΣsuperscriptsubscript𝑥𝑣212\|\xi\|=(\sum_{v\in\Sigma_{\infty}}x_{v}^{2})^{1/2} on Fsubscript𝐹F_{\infty} for estimations.

Lemma 7.18.

For any (t:n)F𝒬FIrr(t:n)_{F}\in{\mathcal{Q}}_{F}^{\rm Irr} with t,n𝔬𝑡𝑛𝔬t,\,n\in\mathfrak{o}, tn0𝑡𝑛0tn\not=0 and a non-zero ideal 𝔞𝔬𝔞𝔬{\mathfrak{a}}\subset\mathfrak{o} such that t24n𝔞superscript𝑡24𝑛𝔞t^{2}-4n\in{\mathfrak{a}}, it holds that (|n|v1t(v))vΣ2dF1/2(N(n1𝔞)1/dF4).\|(\sqrt{|n|_{v}}^{-1}t^{(v)})_{v\in\Sigma_{\infty}}\|^{2}\geqslant d_{F}^{1/2}({\operatorname{N}}(n^{-1}{\mathfrak{a}})^{1/d_{F}}-4).

Proof.

From t24n𝔞superscript𝑡24𝑛𝔞t^{2}-4n\in{\mathfrak{a}}, there is an integral ideal 𝔟𝔟{\mathfrak{b}} such that (t24n)=𝔞𝔟superscript𝑡24𝑛𝔞𝔟(t^{2}-4n)={\mathfrak{a}}{\mathfrak{b}}. By taking the norm, we have |N(t24n)|=N(𝔞)N(𝔟)N(𝔞)Nsuperscript𝑡24𝑛N𝔞N𝔟N𝔞|{\operatorname{N}}(t^{2}-4n)|={\operatorname{N}}({\mathfrak{a}})\,{\operatorname{N}}({\mathfrak{b}})\geqslant{\operatorname{N}}({\mathfrak{a}}) on one hand. On the other hand, by the geometric-arithmetic mean inequality,

|N(t2n14)|=vΣ|t2n14|v{vΣ|t2n14|v2dF}dF/2=dFdF/2ι(t2n14)dF.Nsuperscript𝑡2superscript𝑛14subscriptproduct𝑣subscriptΣsubscriptsuperscript𝑡2superscript𝑛14𝑣superscriptsubscript𝑣subscriptΣsuperscriptsubscriptsuperscript𝑡2superscript𝑛14𝑣2subscript𝑑𝐹subscript𝑑𝐹2superscriptsubscript𝑑𝐹subscript𝑑𝐹2superscriptnormsubscript𝜄superscript𝑡2superscript𝑛14subscript𝑑𝐹\displaystyle|{\operatorname{N}}({t^{2}}{n^{-1}}-4)|=\prod_{v\in\Sigma_{\infty}}|t^{2}n^{-1}-4|_{v}\leqslant\biggl{\{}\sum_{v\in\Sigma_{\infty}}\tfrac{|t^{2}n^{-1}-4|_{v}^{2}}{d_{F}}\biggr{\}}^{d_{F}/2}=d_{F}^{-d_{F}/2}\|\iota_{\infty}(t^{2}n^{-1}-4)\|^{d_{F}}.

Thus ι(t2n14)dF|N(n)|1/dFN(𝔞)1/dF.\|\iota_{\infty}(t^{2}n^{-1}-4)\|\geqslant\sqrt{d_{F}}\,|{\operatorname{N}}(n)|^{-1/d_{F}}{\operatorname{N}}({\mathfrak{a}})^{1/d_{F}}. Set ξ=(|n|v1t(v))vΣ𝜉subscriptsuperscriptsubscript𝑛𝑣1superscript𝑡𝑣𝑣subscriptΣ\xi=(\sqrt{|n|_{v}}^{-1}t^{(v)})_{v\in\Sigma_{\infty}}. Then from the inequality (vxv4)1/2vxv2superscriptsubscript𝑣superscriptsubscript𝑥𝑣412subscript𝑣superscriptsubscript𝑥𝑣2(\sum_{v}x_{v}^{4})^{1/2}\leqslant\sum_{v}x_{v}^{2},

ξ2+4dF1/2superscriptnorm𝜉24superscriptsubscript𝑑𝐹12\displaystyle\|\xi\|^{2}+4d_{F}^{1/2} ι(t2n1)+ι(4)ι(t2n14)dF|N(n)|1/dFN(𝔞)1/dF.\displaystyle\geqslant\|\iota_{\infty}(t^{2}n^{-1})\|+\|\iota_{\infty}(4)\|\geqslant\|\iota_{\infty}(t^{2}n^{-1}-4)\|\geqslant\sqrt{d_{F}}\,|{\operatorname{N}}(n)|^{-1/d_{F}}{\operatorname{N}}({\mathfrak{a}})^{1/d_{F}}.

Set

f(x)=vΣ|xv24|ϱ(z)|Re(z)|4+ϵ2vΣ{(|xvxv241|v+)lv/2|xv|1+|Re(z)|2+ϵ(|xv|>2),(|xv24|v1/2+|xv|)12lv4|xv24|v2lv+2|Re(z)|+1+4ϵ8.(|xv|2)𝑓𝑥subscriptproduct𝑣subscriptΣsuperscriptsuperscriptsubscript𝑥𝑣24italic-ϱ𝑧Re𝑧4italic-ϵ2subscriptproduct𝑣subscriptΣcasessuperscriptsuperscriptsubscriptsubscript𝑥𝑣superscriptsubscript𝑥𝑣241𝑣subscript𝑙𝑣2superscriptsubscript𝑥𝑣1Re𝑧2italic-ϵsubscript𝑥𝑣2superscriptsuperscriptsubscriptsuperscriptsubscript𝑥𝑣24𝑣12subscript𝑥𝑣12subscript𝑙𝑣4superscriptsubscriptsuperscriptsubscript𝑥𝑣24𝑣2subscript𝑙𝑣2Re𝑧14italic-ϵ8subscript𝑥𝑣2f(x)=\prod_{v\in\Sigma_{\infty}}|x_{v}^{2}-4|^{\frac{\varrho(z)-|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}\,\prod_{v\in\Sigma_{\infty}}\begin{cases}\biggl{(}\left|\tfrac{x_{v}}{\sqrt{x_{v}^{2}-4}}-1\right|_{v}^{+}\biggr{)}^{l_{v}/2}|x_{v}|^{\frac{1+|\operatorname{Re}(z)|}{2}+\epsilon}\quad&(|x_{v}|>2),\\ \left(|x_{v}^{2}-4|_{v}^{1/2}+{|x_{v}|}\right)^{\frac{1-2l_{v}}{4}}|x_{v}^{2}-4|_{v}^{\frac{2l_{v}+2|\operatorname{Re}(z)|+1+4\epsilon}{8}}.\quad&(|x_{v}|\leqslant 2)\end{cases}

for x=(xv)dF𝑥subscript𝑥𝑣superscriptsubscript𝑑𝐹x=(x_{v})\in{\mathbb{R}}^{d_{F}}. Set ξ=(|n|v1t(v))vΣ𝜉subscriptsuperscriptsubscript𝑛𝑣1superscript𝑡𝑣𝑣subscriptΣ\xi=(\sqrt{|n|_{v}}^{-1}t^{(v)})_{v\in\Sigma_{\infty}}. From Lemma 7.18, the series (7.12) is bounded by

(7.13) vΣ|n|vϱ(z)|Re(z)|4+ϵ2ξLi(𝔠,n){0}δ(ξdF1/4max(0,N(n1𝔫12𝔠2)1/dF4))f(ξ)\displaystyle\prod_{v\in\Sigma_{\infty}}|n|_{v}^{\frac{\varrho(z)-|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}\sum_{\xi\in L_{i}({\mathfrak{c}},n)-\{0\}}\delta\biggl{(}\|\xi\|\geqslant\sqrt{d_{F}^{1/4}\,\max(0,{\operatorname{N}}(n^{-1}{\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d_{F}}-4)}\biggr{)}\,f(\xi)

where Li(𝔠,n)subscript𝐿𝑖𝔠𝑛L_{i}({\mathfrak{c}},n) denotes the {\mathbb{Z}}-lattice {(Nvxv)vΣ|xι(𝔞i𝔠)}conditional-setsubscriptsubscript𝑁𝑣subscript𝑥𝑣𝑣subscriptΣ𝑥subscript𝜄subscript𝔞𝑖𝔠\{(N_{v}x_{v})_{v\in\Sigma_{\infty}}|\,x\in\iota_{\infty}({\mathfrak{a}}_{i}{\mathfrak{c}})\} in dFsuperscriptsubscript𝑑𝐹{\mathbb{R}}^{d_{F}}. The function f(x)𝑓𝑥f(x) is continuous on dFsuperscriptsubscript𝑑𝐹{\mathbb{R}}^{d_{F}} satisfying the majorization |f(x)|vΣ(1+|xv|)lv+1+ϱ(z)2+2ϵmuch-less-than𝑓𝑥subscriptproduct𝑣subscriptΣsuperscript1subscript𝑥𝑣subscript𝑙𝑣1italic-ϱ𝑧22italic-ϵ|f(x)|\ll\prod_{v\in\Sigma_{\infty}}(1+|x_{v}|)^{-l_{v}+\frac{1+\varrho(z)}{2}+2\epsilon}.

Lemma 7.19.

Let L=(Lj)j=1dF(+)dF𝐿superscriptsubscriptsubscript𝐿𝑗𝑗1subscript𝑑𝐹superscriptsubscriptsubscript𝑑𝐹L=(L_{j})_{j=1}^{d_{F}}\in({\mathbb{R}}_{+})^{d_{F}} and define a function on dFsuperscriptsubscript𝑑𝐹{\mathbb{R}}^{d_{F}} as fL(x)=j=1dF(1+|xj|)Ljsubscript𝑓𝐿𝑥superscriptsubscriptproduct𝑗1subscript𝑑𝐹superscript1subscript𝑥𝑗subscript𝐿𝑗f_{L}(x)=\prod_{j=1}^{d_{F}}(1+|x_{j}|)^{-L_{j}}. Suppose L¯>1¯𝐿1{\underline{L}}>1. Then we have a constant C>0𝐶0C>0 such that for any A0𝐴0A\geqslant 0 and for any pair of {\mathbb{Z}}-lattices ΛΛ0ΛsubscriptΛ0\Lambda\subset\Lambda_{0} of full rank in dFsuperscriptsubscript𝑑𝐹{\mathbb{R}}^{d_{F}}, we have

ξΛ{0}δ(ξA)fL(ξ)Cr(Λ0)dF(1+r(Λ0))dFL¯max(A,r(Λ))1L¯\displaystyle\sum_{\xi\in\Lambda-\{0\}}\delta(\|\xi\|\geqslant A)\,f_{L}(\xi)\leqslant C\,r(\Lambda_{0})^{-d_{F}}(1+r(\Lambda_{0}))^{d_{F}\bar{L}}\,\max(A,r(\Lambda))^{1-{\underline{L}}}

with L¯=min{Lj| 1jdF}¯𝐿conditionalsubscript𝐿𝑗1𝑗subscript𝑑𝐹{\underline{L}}=\min\{L_{j}|\,1\leqslant j\leqslant d_{F}\}, L¯=max{Lj| 1jdF}¯𝐿conditionalsubscript𝐿𝑗1𝑗subscript𝑑𝐹\bar{L}=\max\{L_{j}|\,1\leqslant j\leqslant d_{F}\} and r(Λ)=21infξΛ{0}ξ𝑟Λsuperscript21subscriptinfimum𝜉Λ0norm𝜉r(\Lambda)=2^{-1}\inf_{\xi\in\Lambda-\{0\}}\|\xi\|.

Proof.

This follows from the proof of [27, Theorem A.1]. ∎

Set Λn={(|n|v1xv)|xι(𝔬)}subscriptΛ𝑛conditional-setsuperscriptsubscript𝑛𝑣1subscript𝑥𝑣𝑥subscript𝜄𝔬\Lambda_{n}=\{(\sqrt{|n|_{v}}^{-1}x_{v})|\,x\in\iota_{\infty}(\mathfrak{o})\}. Since Li(𝔠,n)subscript𝐿𝑖𝔠𝑛L_{i}({\mathfrak{c}},n) are all contained in ΛnsubscriptΛ𝑛\Lambda_{n}, from Lemma 7.19, the series (7.13) is absolutely convergent if l¯4¯𝑙4{\underline{l}}\geqslant 4 and ϵ>0italic-ϵ0\epsilon>0 is small enough, and majorized by

vΣ|n|vϱ(z)|Re(z)|4+ϵ2×δ(n)max(dF1/4max(0,N(n1𝔫12𝔠2)1/dF4),r(Li(𝔠,n)))1L¯(z)\displaystyle\prod_{v\in\Sigma_{\infty}}|n|_{v}^{\frac{\varrho(z)-|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}\times\delta(n)\,\max\biggl{(}\sqrt{d_{F}^{1/4}\max(0,{\operatorname{N}}(n^{-1}{\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d_{F}}-4)},\,r(L_{i}({\mathfrak{c}},n))\biggr{)}^{1-{\underline{L}}(z)}

with L¯(z)=l¯1+ϱ(z)22ϵ1+ϵ¯𝐿𝑧¯𝑙1italic-ϱ𝑧22italic-ϵ1italic-ϵ{\underline{L}}(z)={\underline{l}}-\frac{1+\varrho(z)}{2}-2\epsilon\geqslant 1+\epsilon and δ(n)=r(Λn)dF(1+r(Λn))dFL¯(z)𝛿𝑛𝑟superscriptsubscriptΛ𝑛subscript𝑑𝐹superscript1𝑟subscriptΛ𝑛subscript𝑑𝐹¯𝐿𝑧\delta(n)=r(\Lambda_{n})^{-d_{F}}(1+r(\Lambda_{n}))^{d_{F}{\overline{L}}(z)} with L¯(z)=l¯1+ϱ(z)22ϵ¯𝐿𝑧¯𝑙1italic-ϱ𝑧22italic-ϵ\bar{L}(z)=\overline{l}-\frac{1+\varrho(z)}{2}-2\epsilon. By a slight modification of the proof of [27, Lemma A.7], we have |N(n)|1/2dFdF1/2r(Λn)superscriptN𝑛12subscript𝑑𝐹superscriptsubscript𝑑𝐹12𝑟subscriptΛ𝑛|{\operatorname{N}}(n)|^{-1/2d_{F}}\leqslant d_{F}^{1/2}r(\Lambda_{n}) and (|N(n)|1/2N(𝔞i𝔠))1/dFr(Li(𝔠,n))superscriptsuperscriptN𝑛12Nsubscript𝔞𝑖𝔠1subscript𝑑𝐹𝑟subscript𝐿𝑖𝔠𝑛(|{\operatorname{N}}(n)|^{-1/2}{\operatorname{N}}({\mathfrak{a}}_{i}{\mathfrak{c}}))^{1/d_{F}}\leqslant r(L_{i}({\mathfrak{c}},n)). From Minkowski’s convex body theorem, we have an upper-bound r(Λn)|N(n)|1/2dFmuch-less-than𝑟subscriptΛ𝑛superscriptN𝑛12subscript𝑑𝐹r(\Lambda_{n})\ll|{\operatorname{N}}(n)|^{-1/2d_{F}}. Hence

δ(n)|N(n)|1/2(1+|N(n)|1/2dF)dFL¯(z)|N(n)|1/2for n𝔬.formulae-sequencemuch-less-than𝛿𝑛superscriptN𝑛12superscript1superscriptN𝑛12subscript𝑑𝐹subscript𝑑𝐹¯𝐿𝑧much-less-thansuperscriptN𝑛12for n𝔬\delta(n)\ll|{\operatorname{N}}(n)|^{1/2}(1+|{\operatorname{N}}(n)|^{-1/2d_{F}})^{d_{F}\overline{L}(z)}\ll|{\operatorname{N}}(n)|^{1/2}\quad\text{for $n\in\mathfrak{o}$}.

As a consequence, we obtain a majorant of the series (7.13) as

|N(n)|ϱ(z)|Re(z)|4+ϵ2|N(n)|1/2max(dF1/4max(0,N(n1𝔫12𝔠2)1/dF4),|N(n)|1/2dFN(𝔞i𝔠)1/dF)1L¯(z).\displaystyle|{\operatorname{N}}(n)|^{\frac{\varrho(z)-|\operatorname{Re}(z)|}{4}+\frac{\epsilon}{2}}\,|{\operatorname{N}}(n)|^{1/2}\,\max\biggl{(}\sqrt{d_{F}^{1/4}\max(0,{\operatorname{N}}(n^{-1}{\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d_{F}}-4)},|{\operatorname{N}}(n)|^{-1/2d_{F}}{\operatorname{N}}({\mathfrak{a}}_{i}{\mathfrak{c}})^{1/d_{F}}\biggr{)}^{1-{\underline{L}}(z)}.

From the argument so far, we obtain the half of the following majorization when R(n)={0}𝑅𝑛0R(n)=\{0\}.

Lemma 7.20.

Let ϵ>0italic-ϵ0\epsilon>0 be small enough. Let 𝔫𝔫{\mathfrak{n}} be a square-free integral ideal satisfying the assumption (7.11), 𝔠𝔠{\mathfrak{c}} a divisor of 𝔫𝔫{\mathfrak{n}}, ε𝔬×/(𝔬×)2𝜀superscript𝔬superscriptsuperscript𝔬2\varepsilon\in\mathfrak{o}^{\times}/(\mathfrak{o}^{\times})^{2}, iI(S,ν)𝑖𝐼𝑆𝜈i\in I(S,\nu) and ν0S𝜈superscriptsubscript0𝑆\nu\in{\mathbb{N}}_{0}^{S}. Then there exists a constant C>0𝐶0C>0 independent of (𝔫,𝔠,i,ν)𝔫𝔠𝑖𝜈({\mathfrak{n}},{\mathfrak{c}},i,\nu) such that the following inequality holds uniformly in (z,𝐬)×𝔛S𝑧𝐬subscript𝔛𝑆(z,{\mathbf{s}})\in{\mathbb{C}}\times{\mathfrak{X}}_{S} with Re(z)[2l¯+3+6ϵ,l¯1/2ϵ]Re𝑧2¯𝑙36italic-ϵ¯𝑙12italic-ϵ\operatorname{Re}(z)\in[-2\underline{l}+3+6\epsilon,\underline{l}-1/2-\epsilon], minvSRe(sv)>max(1,|Re(z)|+12)subscript𝑣𝑆Resubscript𝑠𝑣1Re𝑧12\min_{v\in S}\operatorname{Re}(s_{v})>\max(1,\tfrac{|\operatorname{Re}(z)|+1}{2}), where L¯(z)=l¯1+ϱ(z)22ϵ¯𝐿𝑧¯𝑙1italic-ϱ𝑧22italic-ϵ{\underline{L}}(z)={\underline{l}}-\frac{1+\varrho(z)}{2}-2\epsilon and ϱ(z)=max(1,|Re(z)|)italic-ϱ𝑧1Re𝑧\varrho(z)=\max(1,|\operatorname{Re}(z)|) :

|Ξ(z,𝐬)(𝔠,𝔫1,i,εni,ν)\displaystyle|\Xi^{(z,{\mathbf{s}})}({\mathfrak{c}},{\mathfrak{n}}_{1},i,\varepsilon n_{i,\nu})
\displaystyle\leqslant CvS(𝔫𝔫11)2qv+1vS(𝔫1)4(1+qv1/2)1+qvvSqvνv(Re(sv)+ϱ(z)4+ϵ)|N(ni,ν)|1/2+(L¯(z)1)/2dF𝐶subscriptproduct𝑣𝑆𝔫superscriptsubscript𝔫112subscript𝑞𝑣1subscriptproduct𝑣𝑆subscript𝔫141superscriptsubscript𝑞𝑣121subscript𝑞𝑣subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣subscript𝜈𝑣Resubscript𝑠𝑣italic-ϱ𝑧4italic-ϵsuperscriptNsubscript𝑛𝑖𝜈12¯𝐿𝑧12subscript𝑑𝐹\displaystyle\,C\,\prod_{v\in S({\mathfrak{n}}{\mathfrak{n}}_{1}^{-1})}\frac{2}{q_{v}+1}\prod_{v\in S({\mathfrak{n}}_{1})}\frac{4(1+q_{v}^{1/2})}{1+q_{v}}\,\prod_{v\in S}q_{v}^{\nu_{v}(\frac{-\operatorname{Re}(s_{v})+\varrho(z)}{4}+\epsilon)}|{\operatorname{N}}(n_{i,\nu})|^{1/2+({\underline{L}}(z)-1)/2d_{F}}
×max(dF1/4max( 0,N(𝔫12𝔠2)1/dF4|N(ni,ν)|1/dF),N(𝔞i𝔠)1/dF)1L¯(z)\displaystyle\times\max\biggl{(}\sqrt{d_{F}^{1/4}\max\bigl{(}\,0,{\operatorname{N}}({\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d_{F}}-4|{\operatorname{N}}(n_{i,\nu})|^{1/d_{F}}\,\,\bigr{)}},\,{\operatorname{N}}({\mathfrak{a}}_{i}{\mathfrak{c}})^{1/d_{F}}\biggr{)}^{1-{\underline{L}}(z)}
+δ(R(εni,ν)=)δ((νv)vS{0,1}S)vS(𝔫)21+qvvSmax(1,qvRe(sv)+|Re(z)|4+ϵ).𝛿𝑅𝜀subscript𝑛𝑖𝜈𝛿subscriptsubscript𝜈𝑣𝑣𝑆superscript01𝑆subscriptproduct𝑣𝑆𝔫21subscript𝑞𝑣subscriptproduct𝑣𝑆1superscriptsubscript𝑞𝑣Resubscript𝑠𝑣Re𝑧4italic-ϵ\displaystyle+\delta(R(\varepsilon n_{i,\nu})=\varnothing)\,\delta((\nu_{v})_{v\in S}\in\{0,1\}^{S})\,\prod_{v\in S({\mathfrak{n}})}\frac{2}{1+q_{v}}\prod_{v\in S}\max(1,q_{v}^{\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\epsilon}).
Proof.

The case R(εni,ν)={0}𝑅𝜀subscript𝑛𝑖𝜈0R(\varepsilon n_{i,\nu})=\{0\} is settled before the lemma. Suppose R(εni,ν)=𝑅𝜀subscript𝑛𝑖𝜈R(\varepsilon n_{i,\nu})=\varnothing. In this case we have to estimate the extra terms with t=0𝑡0t=0 in Ξ(z,𝐬)(𝔠,𝔫1,i,n)superscriptΞ𝑧𝐬𝔠subscript𝔫1𝑖𝑛\Xi^{(z,{\mathbf{s}})}({\mathfrak{c}},{\mathfrak{n}}_{1},i,n). Set n=εni,ν𝑛𝜀subscript𝑛𝑖𝜈n=\varepsilon n_{i,\nu}. By t=0𝑡0t=0, (from Lemma 7.17) we have νv{0,1}(vS)subscript𝜈𝑣01for-all𝑣𝑆\nu_{v}\in\{0,1\}\,(\forall v\in S). Combining this with the relation (4n)(v)=mv2Δv0superscript4𝑛𝑣superscriptsubscript𝑚𝑣2superscriptsubscriptΔ𝑣0(-4n)^{(v)}=m_{v}^{2}\Delta_{v}^{0}, we have mv𝔬v×subscript𝑚𝑣superscriptsubscript𝔬𝑣m_{v}\in\mathfrak{o}_{v}^{\times} for all vS𝑣𝑆v\in S. Thus the S𝑆S-factor is bounded by vS16max(1,qvRe(sv)+|Re(z)|4+ϵ)subscriptproduct𝑣𝑆161superscriptsubscript𝑞𝑣Resubscript𝑠𝑣Re𝑧4italic-ϵ\prod_{v\in S}16\max(1,q_{v}^{\frac{-\operatorname{Re}(s_{v})+|\operatorname{Re}(z)|}{4}+\epsilon}) by Lemma 7.14 (3). The equality R(εni,ν)=𝑅𝜀subscript𝑛𝑖𝜈R(\varepsilon n_{i,\nu})=\varnothing means that n(v)(mod𝔭v)annotatedsuperscript𝑛𝑣pmodsubscript𝔭𝑣n^{(v)}\pmod{{\mathfrak{p}}_{v}} is a square residue at all vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}). Hence n(v)(𝔬v×)2superscript𝑛𝑣superscriptsuperscriptsubscript𝔬𝑣2n^{(v)}\in(\mathfrak{o}_{v}^{\times})^{2} for all vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}). Thus from the case of Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1 in Lemma 7.14 (2), we see vS(𝔫)|𝔈v(z)(γ^v)|vS(𝔫)21+qv.subscriptproduct𝑣𝑆𝔫superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣subscriptproduct𝑣𝑆𝔫21subscript𝑞𝑣\prod_{v\in S({\mathfrak{n}})}|{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})|\leqslant\prod_{v\in S({\mathfrak{n}})}\tfrac{2}{1+q_{v}}. The archimedean component to be accounted for is the product of 𝒪v,(z)(0)=2lv1(1)lv/2πΓ(lv2+z14)Γ(lv2+z14)Γ(lv)1superscriptsubscript𝒪𝑣𝑧0superscript2subscript𝑙𝑣1superscript1subscript𝑙𝑣2𝜋Γsubscript𝑙𝑣2𝑧14Γsubscript𝑙𝑣2𝑧14Γsuperscriptsubscript𝑙𝑣1{\mathcal{O}}_{v}^{-,(z)}(0)=2^{l_{v}-1}(-1)^{l_{v}/2}\sqrt{\pi}\Gamma(\frac{l_{v}}{2}+\frac{z-1}{4})\Gamma(\frac{l_{v}}{2}+\frac{-z-1}{4})\Gamma(l_{v})^{-1}, which is bounded on the vertical strip Re(z)[2l¯+3+6ϵ,l¯1/2ϵ]Re𝑧2¯𝑙36italic-ϵ¯𝑙12italic-ϵ\operatorname{Re}(z)\in[-2\underline{l}+3+6\epsilon,\underline{l}-1/2-\epsilon]. The conditions on Re(z)Re𝑧\operatorname{Re}(z) and minvSRe(sv)subscript𝑣𝑆Resubscript𝑠𝑣\min_{v\in S}\operatorname{Re}(s_{v}) come from L¯(z)>1+ϵ¯𝐿𝑧1italic-ϵ{\underline{L}}(z)>1+\epsilon and Lemma 7.14 (3), (5). ∎

By Lemma 7.20, to complete the proof of Theorem 7.13 it suffices to note the multi-series

ν0SvSqvνv(Re(sv)+ϱ(z)4+ϵ)|N(ni,ν)|1/2+(L¯(z)1)/2dFsubscript𝜈superscriptsubscript0𝑆subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣subscript𝜈𝑣Resubscript𝑠𝑣italic-ϱ𝑧4italic-ϵsuperscriptNsubscript𝑛𝑖𝜈12¯𝐿𝑧12subscript𝑑𝐹\sum_{\nu\in{\mathbb{N}}_{0}^{S}}\prod_{v\in S}q_{v}^{\nu_{v}(\frac{-\operatorname{Re}(s_{v})+\varrho(z)}{4}+\epsilon)}|{\operatorname{N}}(n_{i,\nu})|^{1/2+({\underline{L}}(z)-1)/2d_{F}}

converges absolutely if Re(sv)>ϱ(z)+2dF1(L¯(z)1)+2+4ϵResubscript𝑠𝑣italic-ϱ𝑧2superscriptsubscript𝑑𝐹1¯𝐿𝑧124italic-ϵ\operatorname{Re}(s_{v})>\varrho(z)+2d_{F}^{-1}({\underline{L}}(z)-1)+2+4\epsilon for all vS𝑣𝑆v\in S. Note ϱ(z)+2dF1(L¯(z)1)+4ϵ+2<ϱ(z)+2(L¯(z)1)+4ϵ+2=2l¯1italic-ϱ𝑧2superscriptsubscript𝑑𝐹1¯𝐿𝑧14italic-ϵ2italic-ϱ𝑧2¯𝐿𝑧14italic-ϵ22¯𝑙1\varrho(z)+2d_{F}^{-1}({\underline{L}}(z)-1)+4\epsilon+2<\varrho(z)+2({\underline{L}}(z)-1)+4\epsilon+2=2{\underline{l}}-1 due to L¯(z)>1¯𝐿𝑧1{\underline{L}}(z)>1.

7.5. The conclusion

From (7.5) and (7.6),

(β)Δ(g)=Lσβ(z)ΛF(z+1)EΔ(z;12)vΣFφvΔ,(z)(gv)dz,g=(gv)G𝔸.formulae-sequencesuperscriptsuperscriptsubscript𝛽Δ𝑔subscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧1superscript𝐸Δ𝑧subscript12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝜑𝑣Δ𝑧subscript𝑔𝑣𝑑𝑧𝑔subscript𝑔𝑣subscript𝐺𝔸\displaystyle({\mathcal{E}}_{\beta}^{*})^{\Delta}(g)=\int_{L_{\sigma}}\beta(z)\,\Lambda_{F}(z+1)E^{\Delta}(z;1_{2})\prod_{v\in\Sigma_{F}}\varphi_{v}^{\Delta,(z)}(g_{v})\,{{d}}z,\quad g=(g_{v})\in G_{\mathbb{A}}.

Substituting this to (7.3) and exchanging the order of integrals, we obtain

𝕁ell(𝐬,β)subscript𝕁ell𝐬𝛽\displaystyle{\mathbb{J}}_{\rm{ell}}({\mathbf{s}},\beta) =12γ~𝒬FIrrLσβ(z)ΛF(z+1)EΔ(z;12){𝔗Δ\G𝔸Φ(𝐬;g1γ^g)vΣFφΔ,(z)(gv)dg}𝑑zabsent12subscript~𝛾superscriptsubscript𝒬𝐹Irrsubscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧1superscript𝐸Δ𝑧subscript12subscript\subscript𝔗Δsubscript𝐺𝔸Φ𝐬superscript𝑔1^𝛾𝑔subscriptproduct𝑣subscriptΣ𝐹superscript𝜑Δ𝑧subscript𝑔𝑣𝑑𝑔differential-d𝑧\displaystyle=\tfrac{1}{2}\sum_{\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm{Irr}}}\int_{L_{\sigma}}\beta(z)\,\Lambda_{F}(z+1)E^{\Delta}(z;1_{2})\,\{\int_{{\mathfrak{T}}_{\Delta}\backslash G_{\mathbb{A}}}\Phi({\mathbf{s}};g^{-1}\hat{\gamma}g)\,\prod_{v\in\Sigma_{F}}\varphi^{\Delta,(z)}(g_{v})\,{{d}}g\}\,{{d}}z
=12γ~𝒬FIrrLσβ(z)ΛF(z+1)EΔ(z;12){vΣF𝔈v(z)(γ^v)}𝑑z.absent12subscript~𝛾superscriptsubscript𝒬𝐹Irrsubscriptsubscript𝐿𝜎𝛽𝑧subscriptΛ𝐹𝑧1superscript𝐸Δ𝑧subscript12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣differential-d𝑧\displaystyle=\tfrac{1}{2}\sum_{\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm{Irr}}}\int_{L_{\sigma}}\beta(z)\,\Lambda_{F}(z+1)E^{\Delta}(z;1_{2})\,\{\prod_{v\in\Sigma_{F}}{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})\}\,{{d}}z.

We need to legitimatize the order exchange of integral above. By Fubini’s theorem and Lemma 7.8, it suffices to show the following.

Lemma 7.21.

For a fixed γ~=(t:n)F\tilde{\gamma}=(t:n)_{F}, there exist N>0𝑁0N>0 and a finite set T0ΣFsubscript𝑇0subscriptΣ𝐹T_{0}\subset\Sigma_{F} containing ΣsubscriptΣ\Sigma_{\infty} such that

(7.14) vT𝔗Δ,v\Gv|Φv(gv1γ^vgv)||φvΔ,(z)(gv)|𝑑gv=O((1+|z|)N)subscriptproduct𝑣𝑇subscript\subscript𝔗Δ𝑣subscript𝐺𝑣subscriptΦ𝑣superscriptsubscript𝑔𝑣1subscript^𝛾𝑣subscript𝑔𝑣superscriptsubscript𝜑𝑣Δ𝑧subscript𝑔𝑣differential-dsubscript𝑔𝑣𝑂superscript1𝑧𝑁\displaystyle\prod_{v\in T}\int_{{\mathfrak{T}}_{\Delta,v}\backslash G_{v}}|\Phi_{v}(g_{v}^{-1}\hat{\gamma}_{v}g_{v})|\,|\varphi_{v}^{\Delta,(z)}(g_{v})|\,{{d}}g_{v}=O((1+|z|)^{N})

uniformly for all finite sets TΣF𝑇subscriptΣ𝐹T\subset\Sigma_{F} containing T0subscript𝑇0T_{0} and for all z𝑧z in the strip 2l¯+3+6ϵ|Re(z)|l¯1/2ϵ2¯𝑙36italic-ϵRe𝑧¯𝑙12italic-ϵ-2\underline{l}+3+6\epsilon\leqslant|\operatorname{Re}(z)|\leqslant\underline{l}-1/2-\epsilon with small ϵ>0italic-ϵ0\epsilon>0.

Proof.

For simplicity we argue assuming t0𝑡0t\not=0. (The case t=0𝑡0t=0 is easier.) Let T0subscript𝑇0T_{0} be the union of ΣΣdyadicSS(𝔫DF)subscriptΣsubscriptΣdyadic𝑆𝑆𝔫subscript𝐷𝐹\Sigma_{\infty}\cup\Sigma_{\rm{dyadic}}\cup S\cup S({\mathfrak{n}}D_{F}) and the set of vΣfin𝑣subscriptΣfinv\in\Sigma_{\rm fin} such that |t|v=|4n|v=|t24n|v=1subscript𝑡𝑣subscript4𝑛𝑣subscriptsuperscript𝑡24𝑛𝑣1|t|_{v}=|4n|_{v}=|t^{2}-4n|_{v}=1 does not hold. If vΣfinT0𝑣subscriptΣfinsubscript𝑇0v\in\Sigma_{\rm fin}-T_{0} is such that Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1, then a=t2mv𝑎𝑡2subscript𝑚𝑣a=\frac{t}{2m_{v}}, b=a+1a1𝑏𝑎1𝑎1b=\frac{a+1}{a-1} satisfies |a1|v=|a+1|v=1subscript𝑎1𝑣subscript𝑎1𝑣1|a-1|_{v}=|a+1|_{v}=1 and |b1|v2=|4|v|a21|v=1superscriptsubscript𝑏1𝑣2subscript4𝑣subscriptsuperscript𝑎21𝑣1|b-1|_{v}^{2}=\frac{|4|_{v}}{|a^{2}-1|_{v}}=1. Thus as we have seen in § 7.3,

𝔗Δ,v\Gv|Φv(gv1γ^vgv)||φΔ,(z)(gv)|𝑑gv=subscript\subscript𝔗Δ𝑣subscript𝐺𝑣subscriptΦ𝑣superscriptsubscript𝑔𝑣1subscript^𝛾𝑣subscript𝑔𝑣superscript𝜑Δ𝑧subscript𝑔𝑣differential-dsubscript𝑔𝑣absent\displaystyle\textstyle{\int}_{{\mathfrak{T}}_{\Delta,v}\backslash G_{v}}|\Phi_{v}(g_{v}^{-1}\hat{\gamma}_{v}g_{v})||\varphi^{\Delta,(z)}(g_{v})|\,{{d}}g_{v}= Fv(𝐊v|Φv(k1[b(b1)x01]k)|𝑑k)|φv(0,z)([1x01])|𝑑xsubscriptsubscript𝐹𝑣subscriptsubscript𝐊𝑣subscriptΦ𝑣superscript𝑘1delimited-[]𝑏𝑏1𝑥01𝑘differential-d𝑘superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥\displaystyle\textstyle{\int}_{F_{v}}\left(\int_{{\mathbf{K}}_{v}}|\Phi_{v}\left(k^{-1}\left[\begin{smallmatrix}b&(b-1)x\\ 0&1\end{smallmatrix}\right]k\right)|\,{{d}}k\,\right)|\varphi_{v}^{(0,z)}(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right])|\,{{d}}x
=\displaystyle= x𝔬v|φv(0,z)([1x01])|𝑑x=1.subscript𝑥subscript𝔬𝑣superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥1\displaystyle\textstyle{\int}_{x\in\mathfrak{o}_{v}}\left|\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)\right|\,{{d}}x=1.

If vΣfinT0𝑣subscriptΣfinsubscript𝑇0v\in\Sigma_{\rm fin}-T_{0} is such that Δv0𝔬v×(𝔬v×)2superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}, then since |𝔫v(t,n)|v=|t(v)2mv|v=1subscriptsubscript𝔫𝑣𝑡𝑛𝑣subscriptsuperscript𝑡𝑣2subscript𝑚𝑣𝑣1|{\mathfrak{n}}_{v}(t,n)|_{v}=|\frac{t^{(v)}}{2m_{v}}|_{v}=1, from the proof given in § 10.2.2, we also have 𝔗Δ,v\Gv|Φv(gv1γ^vgv)||φΔ,(z)(gv)|𝑑gv=1subscript\subscript𝔗Δ𝑣subscript𝐺𝑣subscriptΦ𝑣superscriptsubscript𝑔𝑣1subscript^𝛾𝑣subscript𝑔𝑣superscript𝜑Δ𝑧subscript𝑔𝑣differential-dsubscript𝑔𝑣1\int_{{\mathfrak{T}}_{\Delta,v}\backslash G_{v}}|\Phi_{v}(g_{v}^{-1}\hat{\gamma}_{v}g_{v})||\varphi^{\Delta,(z)}(g_{v})|\,{{d}}g_{v}=1. Thus the left-hand side of (7.14) is independent of T𝑇T containing T0subscript𝑇0T_{0}. At places in T0subscript𝑇0T_{0}, the absolute convergence of the integral and their necessary polynomial bound are shown or are easily deducible from the arguments in § 6.3 and in § 10. ∎

Set

(7.15) J^ell(𝐬,z)=12γ~𝒬FIrr𝒬FSΛF(z+1)EΔ(z;12){vΣF𝔈v(z)(γ^v)}.subscript^𝐽ell𝐬𝑧12subscript~𝛾superscriptsubscript𝒬𝐹Irrsuperscriptsubscript𝒬𝐹𝑆subscriptΛ𝐹𝑧1superscript𝐸Δ𝑧subscript12subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣\displaystyle\hat{J}_{\rm{ell}}({\mathbf{s}},z)=\tfrac{1}{2}\sum_{\tilde{\gamma}\in{\mathcal{Q}}_{F}^{\rm{Irr}}\cap{\mathcal{Q}}_{F}^{S}}\Lambda_{F}(z+1)E^{\Delta}(z;1_{2})\,\{\prod_{v\in\Sigma_{F}}{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})\}.

Now we obtain Theorem 4.1 for F𝐹F-elliptic terms in the following form.

Theorem 7.22.

For (z,𝐬)×𝔛S𝑧𝐬subscript𝔛𝑆(z,{\mathbf{s}})\in{\mathbb{C}}\times{\mathfrak{X}}_{S} satisfying z±1𝑧plus-or-minus1z\neq\pm 1, Re(z)(2l¯+3,l¯1/2)Re𝑧2¯𝑙3¯𝑙12\operatorname{Re}(z)\in(-2\underline{l}+3,\underline{l}-1/2), minvSRe(sv)>2l¯1subscript𝑣𝑆Resubscript𝑠𝑣2¯𝑙1\min_{v\in S}\operatorname{Re}(s_{v})>2{\underline{l}}-1, the series (7.15) converges absolutely and locally uniformly in (z,𝐬)𝑧𝐬(z,{\mathbf{s}}). For a fixed 𝐬𝔛S𝐬subscript𝔛𝑆{\mathbf{s}}\in{\mathfrak{X}}_{S} such that minvSRe(sv)>2l¯1subscript𝑣𝑆Resubscript𝑠𝑣2¯𝑙1\min_{v\in S}\operatorname{Re}(s_{v})>2{\underline{l}}-1, the function zJ^ell(𝐬,z)maps-to𝑧subscript^𝐽ell𝐬𝑧z\mapsto\hat{J}_{\rm{ell}}({\mathbf{s}},z) is holomorphic away from z=±1𝑧plus-or-minus1z=\pm 1 and is vertically of moderate growth on the strip Re(z)[2l¯+3+ϵ,l¯1/2ϵ]Re𝑧2¯𝑙3italic-ϵ¯𝑙12italic-ϵ\operatorname{Re}(z)\in[-2{\underline{l}}+3+\epsilon,{\underline{l}}-1/2-\epsilon] for any sufficiently small ϵ>0italic-ϵ0\epsilon>0. We have the formula (4.7) with =ellell\natural={\rm ell}.

8. Proof of the main theorems

For any holomorphic function f(𝐬)𝑓𝐬f({\mathbf{s}}) on 𝔛Ssubscript𝔛𝑆{\mathfrak{X}}_{S} and for any 𝐜=(cv)vSS𝐜subscriptsubscript𝑐𝑣𝑣𝑆superscript𝑆{\mathbf{c}}=(c_{v})_{v\in S}\in{\mathbb{R}}^{S}, the multi-dimensional contour integral L(𝐜)f(𝐬)𝑑μS(𝐬)subscript𝐿𝐜𝑓𝐬differential-dsubscript𝜇𝑆𝐬\int_{L({\mathbf{c}})}f({\mathbf{s}}){{d}}\mu_{S}({\mathbf{s}}) is defined as the iteration of the one-dimensional contour integrals Lv(cv)𝑑μv(j)(sv(j))subscriptsubscript𝐿𝑣subscript𝑐𝑣differential-dsubscript𝜇𝑣𝑗subscript𝑠𝑣𝑗\int_{L_{v}(c_{v})}{{d}}\mu_{v(j)}(s_{v(j)}) (j=1,,k)𝑗1𝑘(j=1,\dots,k) in any ordering S={v(1),,v(k)}𝑆𝑣1𝑣𝑘S=\{v(1),\dots,v(k)\}.

In this section, we let the square-free ideal 𝔫𝔫{\mathfrak{n}} vary in such a way that 𝔫𝔫{\mathfrak{n}} is prime to 2𝔭Sj=1h𝔞j2subscript𝔭𝑆superscriptsubscriptproduct𝑗1subscript𝔞𝑗2{\mathfrak{p}}_{S}\prod_{j=1}^{h}{\mathfrak{a}}_{j}, where 𝔞jsubscript𝔞𝑗{\mathfrak{a}}_{j} are the ideals fixed in § 7 and we set 𝔭S=vS(𝔭v𝔬)subscript𝔭𝑆subscriptproduct𝑣𝑆subscript𝔭𝑣𝔬{\mathfrak{p}}_{S}=\prod_{v\in S}({\mathfrak{p}}_{v}\cap{\mathfrak{o}}).

8.1. Proof of Theorem 1.1 and Corollary 1.2

For {unip,hyp,ell}uniphypell\natural\in\{{\rm unip},{\rm hyp},{\rm ell}\}, we define J^0(𝐬,z)superscriptsubscript^𝐽0𝐬𝑧\hat{J}_{\natural}^{0}({\mathbf{s}},z) by (5.2) for =unipunip\natural={\rm unip} and J^0(𝐬,z)=ζF(z+12)1J^(𝐬,z)superscriptsubscript^𝐽0𝐬𝑧subscript𝜁𝐹superscript𝑧121subscript^𝐽𝐬𝑧\hat{J}_{\natural}^{0}({\mathbf{s}},z)=\zeta_{F}\left(\tfrac{z+1}{2}\right)^{-1}\hat{J}_{\natural}({\mathbf{s}},z) for {hyp,ell}hypell\natural\in\{{\rm hyp},{\rm ell}\}. Set I^cusp0(𝐬,z)=ζF(z+12)1I^cusp(𝐬,z)superscriptsubscript^𝐼cusp0𝐬𝑧subscript𝜁𝐹superscript𝑧121subscript^𝐼cusp𝐬𝑧\hat{I}_{\rm cusp}^{0}({\mathbf{s}},z)=\zeta_{F}\left(\tfrac{z+1}{2}\right)^{-1}\hat{I}_{\rm cusp}({\mathbf{s}},z). Recall the function I^cusp(𝐬,z)subscript^𝐼cusp𝐬𝑧\hat{I}_{\rm cusp}({\mathbf{s}},z) defined by (3.6) and the explicit formulas of J^0(𝐬,z)superscriptsubscript^𝐽0𝐬𝑧\hat{J}_{\natural}^{0}({\mathbf{s}},z) stated in §5, §6 and §7. From §4, we have the identity

I^cusp0(𝐬,z)=DFz4{J^unip0(𝐬,z)+J^unip0(𝐬,z)}+J^hyp0(𝐬,z)+J^ell0(𝐬,z)superscriptsubscript^𝐼cusp0𝐬𝑧superscriptsubscript𝐷𝐹𝑧4subscriptsuperscript^𝐽0unip𝐬𝑧subscriptsuperscript^𝐽0unip𝐬𝑧superscriptsubscript^𝐽hyp0𝐬𝑧superscriptsubscript^𝐽ell0𝐬𝑧\hat{I}_{\rm cusp}^{0}({\mathbf{s}},z)=D_{F}^{\frac{z}{4}}\{\hat{J}^{0}_{\rm unip}({\mathbf{s}},z)+\hat{J}^{0}_{\rm unip}({\mathbf{s}},-z)\}+\hat{J}_{\rm hyp}^{0}({\mathbf{s}},z)+\hat{J}_{\rm ell}^{0}({\mathbf{s}},z)

for |Re(z)|<l¯3Re𝑧¯𝑙3|\operatorname{Re}(z)|<\underline{l}-3 and 𝐬𝔛S𝐬subscript𝔛𝑆{\mathbf{s}}\in{\mathfrak{X}}_{S} with minvSRe(sv)>2l¯1subscript𝑣𝑆Resubscript𝑠𝑣2¯𝑙1\min_{v\in S}\operatorname{Re}(s_{v})>2\underline{l}-1 (cf. Proposition 6.13 and Theorem 7.22). To obtain the formula of J^ell0(𝐬,z)superscriptsubscript^𝐽ell0𝐬𝑧\hat{J}_{\rm ell}^{0}({\mathbf{s}},z) in Theorem 1.1, we examine local conditions on (t,n)𝑡𝑛(t,n) posed by various δ𝛿\delta-factors in Theorem 7.9. From Lemma 7.10, when |Δv0|v=1subscriptsuperscriptsubscriptΔ𝑣0𝑣1|\Delta_{v}^{0}|_{v}=1 we have that |t2mv|v1subscript𝑡2subscript𝑚𝑣𝑣1|\tfrac{t}{2m_{v}}|_{v}\not=1 if and only if |nmv2|v1subscript𝑛superscriptsubscript𝑚𝑣2𝑣1|\tfrac{n}{m_{v}^{2}}|_{v}\geqslant 1; hence t2mv𝔬v×𝑡2subscript𝑚𝑣superscriptsubscript𝔬𝑣\tfrac{t}{2m_{v}}\not\in\mathfrak{o}_{v}^{\times} implies |n4mv2|v|4|v11subscript𝑛4superscriptsubscript𝑚𝑣2𝑣superscriptsubscript4𝑣11|\tfrac{n}{4m_{v}^{2}}|_{v}\geqslant|4|_{v}^{-1}\geqslant 1. Thus the δ𝛿\delta-symbol in the first formula of Theorem 7.9 (2) is simplified to δ(n4mv2𝔭v)𝛿𝑛4superscriptsubscript𝑚𝑣2subscript𝔭𝑣\delta(\tfrac{n}{4m_{v}^{2}}\not\in{\mathfrak{p}}_{v}). By a similar argument, the local conditions posed by δ𝛿\delta-symbols in Theorem 7.9 are reduced to the following:

  • If Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1, then n4mv2𝔭v𝑛4superscriptsubscript𝑚𝑣2subscript𝔭𝑣\tfrac{n}{4m_{v}^{2}}\not\in{\mathfrak{p}}_{v},

  • If Δv0𝔬v×(𝔬v×)2superscriptsubscriptΔ𝑣0superscriptsubscript𝔬𝑣superscriptsuperscriptsubscript𝔬𝑣2\Delta_{v}^{0}\in\mathfrak{o}_{v}^{\times}-(\mathfrak{o}_{v}^{\times})^{2}, vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}), then nmv2𝔬v𝑛superscriptsubscript𝑚𝑣2subscript𝔬𝑣\tfrac{n}{m_{v}^{2}}\not\in\mathfrak{o}_{v},

  • If Δv0𝔭v𝔭v2superscriptsubscriptΔ𝑣0subscript𝔭𝑣superscriptsubscript𝔭𝑣2\Delta_{v}^{0}\in{\mathfrak{p}}_{v}-{\mathfrak{p}}_{v}^{2}, or vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}, Δv0{1,5}superscriptsubscriptΔ𝑣015\Delta_{v}^{0}\in\{-1,-5\}, then nmv2𝔭v𝑛superscriptsubscript𝑚𝑣2subscript𝔭𝑣\tfrac{n}{m_{v}^{2}}\not\in{\mathfrak{p}}_{v}.

We remark that 𝔣Δ𝔬vsubscript𝔣Δsubscript𝔬𝑣{\mathfrak{f}}_{\Delta}\mathfrak{o}_{v} coincides with mv𝔬vsubscript𝑚𝑣subscript𝔬𝑣m_{v}\mathfrak{o}_{v} unless Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1 or vΣdyadic,Δv0=5formulae-sequence𝑣subscriptΣdyadicsuperscriptsubscriptΔ𝑣05v\in\Sigma_{\rm dyadic},\Delta_{v}^{0}=5, in which case it equals 2mv𝔬v2subscript𝑚𝑣subscript𝔬𝑣2m_{v}\mathfrak{o}_{v}. By this, we can write the condition above as follows :

  • vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}) and εΔ,vsubscript𝜀Δ𝑣\varepsilon_{\Delta,v} is unramified and non-trivial, then ordv(n𝔣Δ2)<0subscriptord𝑣𝑛superscriptsubscript𝔣Δ20\operatorname{ord}_{v}(n{\mathfrak{f}}_{\Delta}^{-2})<0,

  • εΔ,v=1subscript𝜀Δ𝑣1\varepsilon_{\Delta,v}=1 or vS(𝔡F(Δ)/F)𝑣𝑆subscript𝔡𝐹Δ𝐹v\in S({\mathfrak{d}}_{F(\sqrt{\Delta})/F}), then ordv(n𝔣Δ2)0subscriptord𝑣𝑛superscriptsubscript𝔣Δ20\operatorname{ord}_{v}(n{\mathfrak{f}}_{\Delta}^{-2})\leqslant 0.

The second condition follows from (t:n)F𝒬FS(t:n)_{F}\in{\mathcal{Q}}_{F}^{S}. Indeed, from the relations cvt𝔬vsubscript𝑐𝑣𝑡subscript𝔬𝑣c_{v}t\in\mathfrak{o}_{v}, cv2n𝔬v×subscriptsuperscript𝑐2𝑣𝑛superscriptsubscript𝔬𝑣c^{2}_{v}n\in\mathfrak{o}_{v}^{\times} and (cvt)24(cvn)=(2mvcv)2Δv0superscriptsubscript𝑐𝑣𝑡24subscript𝑐𝑣𝑛superscript2subscript𝑚𝑣subscript𝑐𝑣2superscriptsubscriptΔ𝑣0(c_{v}t)^{2}-4(c_{v}n)=(2m_{v}c_{v})^{2}\Delta_{v}^{0}, we see mvcv𝔬vsubscript𝑚𝑣subscript𝑐𝑣subscript𝔬𝑣m_{v}c_{v}\in\mathfrak{o}_{v}. Hence ordv(n𝔣Δ2)=ordv(cv2ncv2𝔣Δ2)0subscriptord𝑣𝑛superscriptsubscript𝔣Δ2subscriptord𝑣superscriptsubscript𝑐𝑣2𝑛superscriptsubscript𝑐𝑣2superscriptsubscript𝔣Δ20\operatorname{ord}_{v}({n}{\mathfrak{f}}_{\Delta}^{-2})=\operatorname{ord}_{v}(\frac{c_{v}^{2}n}{c_{v}^{2}{\mathfrak{f}}_{\Delta}^{2}})\leqslant 0. This completes the proof of Theorem 1.1.

Take any α𝒜S𝛼subscript𝒜𝑆\alpha\in{\mathcal{A}}_{S}. For {unip,hyp,ell}uniphypell\natural\in\{{\rm unip},{\rm hyp},{\rm ell}\}, set

𝕁0(𝔫|α,z)=(12πi)#SL(𝐜)J^0(𝐬,z)α(𝐬)𝑑μS(𝐬)superscriptsubscript𝕁0conditional𝔫𝛼𝑧superscript12𝜋𝑖#𝑆subscript𝐿𝐜superscriptsubscript^𝐽0𝐬𝑧𝛼𝐬differential-dsubscript𝜇𝑆𝐬\displaystyle{\mathbb{J}}_{\natural}^{0}({\mathfrak{n}}|\alpha,z)=\left(\tfrac{1}{2\pi i}\right)^{\#S}\int_{L({\mathbf{c}})}\hat{J}_{\natural}^{0}({\mathbf{s}},z)\,\alpha({\mathbf{s}})\,{{d}}\mu_{S}({\mathbf{s}})

with 𝐜=(cv)S𝐜subscript𝑐𝑣superscript𝑆{\mathbf{c}}=(c_{v})\in{\mathbb{R}}^{S}, cv1much-greater-thansubscript𝑐𝑣1c_{v}\gg 1. We note that the ideal 𝔫𝔫{\mathfrak{n}} is implicit in the definition of J^0(𝐬,z)superscriptsubscript^𝐽0𝐬𝑧\hat{J}_{\natural}^{0}({\mathbf{s}},z). Set

𝕀cusp0(𝔫|α,z)=(1)#SC(l,𝔫)1(12πi)#SL(𝐜)I^cusp0(𝐬,z)α(𝐬)𝑑μS(𝐬)superscriptsubscript𝕀cusp0conditional𝔫𝛼𝑧superscript1#𝑆𝐶superscript𝑙𝔫1superscript12𝜋𝑖#𝑆subscript𝐿𝐜superscriptsubscript^𝐼cusp0𝐬𝑧𝛼𝐬differential-dsubscript𝜇𝑆𝐬\mathbb{I}_{\rm cusp}^{0}({\mathfrak{n}}|\alpha,z)=(-1)^{\#S}C(l,{\mathfrak{n}})^{-1}\,\left(\tfrac{1}{2\pi i}\right)^{\#S}\int_{L({\mathbf{c}})}\hat{I}_{\rm cusp}^{0}({\mathbf{s}},z)\,\alpha({\mathbf{s}})\,{{d}}\mu_{S}({\mathbf{s}})

for α(𝐬)𝛼𝐬\alpha({\mathbf{s}}) as above. By taking a multi-dimensional contour integral, we obtain the identity

(8.1) (1)#SC(l,𝔫)𝕀cusp0(𝔫|α,z)=DFz4{𝕁unip0(𝔫|α,z)+𝕁unip0(𝔫|α,z)}+𝕁hyp0(𝔫|α,z)+𝕁ell0(𝔫|α,z).superscript1#𝑆𝐶𝑙𝔫subscriptsuperscript𝕀0cuspconditional𝔫𝛼𝑧superscriptsubscript𝐷𝐹𝑧4subscriptsuperscript𝕁0unipconditional𝔫𝛼𝑧superscriptsubscript𝕁unip0conditional𝔫𝛼𝑧superscriptsubscript𝕁hyp0conditional𝔫𝛼𝑧superscriptsubscript𝕁ell0conditional𝔫𝛼𝑧\displaystyle(-1)^{\#S}C(l,{\mathfrak{n}})\,\mathbb{I}^{0}_{\rm cusp}({\mathfrak{n}}|\alpha,z)=D_{F}^{\frac{z}{4}}\{{\mathbb{J}}^{0}_{\rm unip}({\mathfrak{n}}|\alpha,z)+{\mathbb{J}}_{\rm unip}^{0}({\mathfrak{n}}|\alpha,-z)\}+{\mathbb{J}}_{\rm hyp}^{0}({\mathfrak{n}}|\alpha,z)+{\mathbb{J}}_{\rm ell}^{0}({\mathfrak{n}}|\alpha,z).

from Theorem 1.1. By (3.6) and the formula

12πiLv(c){qv(1+ν)/2+qv(1ν)/2qv(1+s)/2qv(1s)/2}1α(s)𝑑μv(s)=α(ν),12𝜋𝑖subscriptsubscript𝐿𝑣𝑐superscriptsuperscriptsubscript𝑞𝑣1𝜈2superscriptsubscript𝑞𝑣1𝜈2superscriptsubscript𝑞𝑣1𝑠2superscriptsubscript𝑞𝑣1𝑠21𝛼𝑠differential-dsubscript𝜇𝑣𝑠𝛼𝜈\tfrac{1}{2\pi i}\textstyle{\int}_{L_{v}(c)}\{q_{v}^{(1+\nu)/2}+q_{v}^{(1-\nu)/2}-q_{v}^{(1+s)/2}-q_{v}^{(1-s)/2}\}^{-1}\alpha(s){{d}}\mu_{v}(s)=-\alpha(\nu),

we see that the left-hand side of (8.1) coincides with (1.6) multiplied by (1)#SC(l,𝔫)superscript1#𝑆𝐶𝑙𝔫(-1)^{\#S}C(l,{\mathfrak{n}}). This completes the proof of Corollary 1.2.

8.2. Proof of Theorem 1.3

Let z(0,1]𝑧01z\in(0,1]. Then the factor vS(𝔫)(1+qv(z+1)/2)/(1+qv)subscriptproduct𝑣𝑆𝔫1superscriptsubscript𝑞𝑣𝑧121subscript𝑞𝑣\prod_{v\in S({\mathfrak{n}})}(1+q_{v}^{(z+1)/2})/(1+q_{v}) of 𝕁unip0(𝔫|α,z)superscriptsubscript𝕁unip0conditional𝔫𝛼𝑧{\mathbb{J}}_{\rm unip}^{0}({\mathfrak{n}}|\alpha,z) is bounded in absolute value by vS(𝔫)qv(z1)/2=N(𝔫)(z1)/2\prod_{v\in S({\mathfrak{n}})}q_{v}^{(z-1)/2}={\operatorname{N}}({\mathfrak{n}})^{(z-1)/2} from below. The value of 𝕁unip0(𝔫|α,z)+𝕁unip0(𝔫|α,z)superscriptsubscript𝕁unip0conditional𝔫𝛼𝑧superscriptsubscript𝕁unip0conditional𝔫𝛼𝑧{\mathbb{J}}_{\rm unip}^{0}({\mathfrak{n}}|\alpha,z)+{\mathbb{J}}_{\rm unip}^{0}({\mathfrak{n}}|\alpha,-z) at z=0𝑧0z=0 is of the form

(vS(𝔫)1+qv121+qv)×(C0+C1vS(𝔫)logqv1+qv1/2)subscriptproduct𝑣𝑆𝔫1superscriptsubscript𝑞𝑣121subscript𝑞𝑣subscript𝐶0subscript𝐶1subscript𝑣𝑆𝔫subscript𝑞𝑣1superscriptsubscript𝑞𝑣12\left(\prod_{v\in S({\mathfrak{n}})}\frac{1+q_{v}^{\frac{1}{2}}}{1+q_{v}}\right)\times\biggl{(}C_{0}+C_{1}\sum_{v\in S({\mathfrak{n}})}\frac{\log q_{v}}{1+q_{v}^{-1/2}}\biggr{)}

with some quantities C0,C1subscript𝐶0subscript𝐶1C_{0},C_{1} independent of 𝔫𝔫{\mathfrak{n}}. This is bounded from below by N(𝔫)1/2logN(𝔫){\operatorname{N}}({\mathfrak{n}})^{-1/2}\log{\operatorname{N}}({\mathfrak{n}}). Hence to prove Theorem 1.3, it suffices to show the hyperbolic and the elliptic terms are bounded from above by O(N(𝔫)1/2η)O({\operatorname{N}}({\mathfrak{n}})^{-1/2-\eta}) with some η>0𝜂0\eta>0 uniformly on the strip |Re(z)|1Re𝑧1|\operatorname{Re}(z)|\leqslant 1.

Let vS𝑣𝑆v\in S and svsubscript𝑠𝑣s_{v} a complex variable. Let 𝒜v0superscriptsubscript𝒜𝑣0{\mathcal{A}}_{v}^{0} be the space of all Laurent polynomials in ζ=qvsv/2𝜁superscriptsubscript𝑞𝑣subscript𝑠𝑣2\zeta=q_{v}^{-s_{v}/2} which is invariant by the substitution ζζ1𝜁superscript𝜁1\zeta\rightarrow\zeta^{-1}. Set σn(ζ)=ζn+ζnsubscript𝜎𝑛𝜁superscript𝜁𝑛superscript𝜁𝑛\sigma_{n}(\zeta)=\zeta^{n}+\zeta^{-n} for n0𝑛subscript0n\in{\mathbb{N}}_{0}. Then as is well known, the elements σn(ζ)(n0)subscript𝜎𝑛𝜁𝑛subscript0\sigma_{n}(\zeta)\,(n\in{\mathbb{N}}_{0}) form a {\mathbb{C}}-basis of the space 𝒜v0superscriptsubscript𝒜𝑣0{\mathcal{A}}_{v}^{0}. For m0𝑚subscript0m\in{\mathbb{N}}_{0}, let 𝒜v0[m]superscriptsubscript𝒜𝑣0delimited-[]𝑚{\mathcal{A}}_{v}^{0}[m] be the {\mathbb{C}}-linear span of functions σn(0nm)subscript𝜎𝑛0𝑛𝑚\sigma_{n}(0\leqslant n\leqslant m). For an integral ideal 𝔞𝔞{\mathfrak{a}} with the prime decomposition 𝔞=vS(𝔭v𝔬)nv𝔞subscriptproduct𝑣𝑆superscriptsubscript𝔭𝑣𝔬subscript𝑛𝑣{\mathfrak{a}}=\prod_{v\in S}({\mathfrak{p}}_{v}\cap{\mathfrak{o}})^{n_{v}} with nvsubscript𝑛𝑣n_{v}\in{\mathbb{N}}, set 𝒜(𝔞)=vS(𝔞)𝒜v0[nv]𝒜𝔞subscripttensor-product𝑣𝑆𝔞superscriptsubscript𝒜𝑣0delimited-[]subscript𝑛𝑣{\mathcal{A}}({\mathfrak{a}})=\bigotimes_{v\in S({\mathfrak{a}})}{\mathcal{A}}_{v}^{0}[n_{v}].

Lemma 8.1.

Let m0𝑚subscript0m\in{\mathbb{N}}_{0}. We have 𝒮^vδ,(z)(α;a)=0superscriptsubscript^𝒮𝑣𝛿𝑧𝛼𝑎0\hat{\mathcal{S}}_{v}^{\delta,(z)}(\alpha;a)=0 for all α𝒜v0[m]𝛼superscriptsubscript𝒜𝑣0delimited-[]𝑚\alpha\in{\mathcal{A}}_{v}^{0}[m] if ordv(a)>msubscriptord𝑣𝑎𝑚\operatorname{ord}_{v}(a)>m.

Proof.

Let e:=ordv(a)>massign𝑒subscriptord𝑣𝑎𝑚e:=\operatorname{ord}_{v}(a)>m. Then from (1.2), we have that 𝒮^vδ,(z)(σn,a)superscriptsubscript^𝒮𝑣𝛿𝑧subscript𝜎𝑛𝑎\hat{\mathcal{S}}_{v}^{\delta,(z)}(\sigma_{n},a) equals

qve/212πi|ζ|=qc/2(1εδ(ϖv)qv1ζ2)(1ζ2)ζen(1+ζ2n)(1q(z+1)/2ζ2)(1qv(z1)/2ζ2)dζζ,superscriptsubscript𝑞𝑣𝑒212𝜋𝑖subscriptcontour-integral𝜁superscript𝑞𝑐21subscript𝜀𝛿subscriptitalic-ϖ𝑣superscriptsubscript𝑞𝑣1superscript𝜁21superscript𝜁2superscript𝜁𝑒𝑛1superscript𝜁2𝑛1superscript𝑞𝑧12superscript𝜁21superscriptsubscript𝑞𝑣𝑧12superscript𝜁2𝑑𝜁𝜁\displaystyle q_{v}^{-e/2}\frac{1}{2\pi i}{\textstyle{\oint}}_{|\zeta|=q^{-c/2}}\frac{(1-\varepsilon_{\delta}(\varpi_{v})\,q_{v}^{-1}\zeta^{2})(1-\zeta^{2})\zeta^{e-n}(1+\zeta^{2n})}{(1-q^{-(z+1)/2}\zeta^{2})(1-q_{v}^{(z-1)/2}\zeta^{2})}\frac{{{d}}\zeta}{\zeta},

which is zero by Cauchy’s theorem if en10𝑒𝑛10e-n-1\geqslant 0. ∎

Let us take a test function α𝒜(𝔞)𝛼𝒜𝔞\alpha\in{\mathcal{A}}({\mathfrak{a}}).

Lemma 8.2.

Let σ(0,1]𝜎01\sigma\in(0,1]. Suppose l𝑙l satisfies l¯>σ+3¯𝑙𝜎3{\underline{l}}>\sigma+3. Set δ=1𝛿1\delta=1 if l¯>dF+3+σ¯𝑙subscript𝑑𝐹3𝜎{\underline{l}}>d_{F}+3+\sigma and δ=1/2+(l¯3σ)/(2dF)𝛿12¯𝑙3𝜎2subscript𝑑𝐹\delta=1/2+(\underline{l}-3-\sigma)/(2d_{F}) otherwise. Then 1/2<δ112𝛿11/2<\delta\leqslant 1, and for any small ϵ>0italic-ϵ0\epsilon>0,

(8.2) |𝕁hyp0(𝔫|α,z)|\displaystyle|{\mathbb{J}}_{\rm hyp}^{0}({\mathfrak{n}}|\alpha,z)|\ll N(𝔫)δ+ϵ\displaystyle{\operatorname{N}}({\mathfrak{n}})^{-\delta+\epsilon}

uniformly for z𝑧z on the strip |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma.

Proof.

From (6.13) and Theorem 6.4 (2) and (3),

(8.3) 𝕁hyp0(𝔫|α,z)superscriptsubscript𝕁hyp0conditional𝔫𝛼𝑧\displaystyle{\mathbb{J}}_{\rm hyp}^{0}({\mathfrak{n}}|\alpha,z) =12ζF,fin(1z2)a𝔬(S)×{1}vΣfinS𝔉v(z)(a)×vΣ𝔉~v(z)(a)absent12subscript𝜁𝐹fin1𝑧2subscript𝑎𝔬superscript𝑆1subscriptproduct𝑣subscriptΣfin𝑆superscriptsubscript𝔉𝑣𝑧𝑎subscriptproduct𝑣subscriptΣsuperscriptsubscript~𝔉𝑣𝑧𝑎\displaystyle=\tfrac{1}{2}\zeta_{F,{\rm fin}}\left(\tfrac{1-z}{2}\right)\sum_{a\in\mathfrak{o}(S)^{\times}-\{1\}}\prod_{v\in\Sigma_{\rm fin}-S}{\mathfrak{F}}_{v}^{(z)}(a)\times\prod_{v\in\Sigma_{\infty}}\tilde{\mathfrak{F}}_{v}^{(z)}(a)
(8.4) ×vS12πiLv(cv)𝔉v(z)(sv,a)αv(sv)dμv(sv),\displaystyle\quad\times\prod_{v\in S}\tfrac{1}{2\pi i}\textstyle{\int}_{L_{v}(c_{v})}{\mathfrak{F}}_{v}^{(z)}(s_{v},a)\,\alpha_{v}(s_{v})\,{{d}}\mu_{v}(s_{v}),

where S=S(𝔞)𝑆𝑆𝔞S=S({\mathfrak{a}}) and 𝔬(S)𝔬𝑆\mathfrak{o}(S) denotes the S𝑆S-integer ring of F𝐹F, and we set 𝔉~v(z)(a)=Γ(1z2)𝔉v(z)(a)superscriptsubscript~𝔉𝑣𝑧𝑎subscriptΓ1𝑧2superscriptsubscript𝔉𝑣𝑧𝑎\tilde{\mathfrak{F}}_{v}^{(z)}(a)=\Gamma_{{\mathbb{R}}}(\tfrac{1-z}{2}){\mathfrak{F}}_{v}^{(z)}(a) for vΣ𝑣subscriptΣv\in\Sigma_{\infty}. By Theorem 6.4 (4) and Lemma 8.1, the S𝑆S-factor (8.4) vanishes unless a(v)𝔭vnvsuperscript𝑎𝑣superscriptsubscript𝔭𝑣subscript𝑛𝑣a^{(v)}\in{\mathfrak{p}}_{v}^{-n_{v}} for all vS𝑣𝑆v\in S. Hence we may suppose that the summation in (8.3) is over the set a𝔞1{0,1}𝑎superscript𝔞101a\in{\mathfrak{a}}^{-1}-\{0,1\}. From Corollary 6.9, the v𝑣v-factor in (8.4) for a(v)𝔭vnvsuperscript𝑎𝑣superscriptsubscript𝔭𝑣subscript𝑛𝑣a^{(v)}\in{\mathfrak{p}}_{v}^{-n_{v}} is majorized by O(|1a|v|Re(z)|+12ϵ)𝑂superscriptsubscript1𝑎𝑣Re𝑧12italic-ϵO(|1-a|_{v}^{-\frac{|\operatorname{Re}(z)|+1}{2}-\epsilon}) uniformly in z𝑧z. Hence as in the proof of Proposition 6.12, we have that for any small ϵ>0italic-ϵ0\epsilon>0,

vΣfinS|𝔉v(z)(a)|vΣ|𝔉~v(z)(a)|vS|12πiLv(cv)𝔉v(z)(sv,a)αv(sv)𝑑μv(sv)|f(a),a𝔞1{0,1}formulae-sequencemuch-less-thansubscriptproduct𝑣subscriptΣfin𝑆superscriptsubscript𝔉𝑣𝑧𝑎subscriptproduct𝑣subscriptΣsuperscriptsubscript~𝔉𝑣𝑧𝑎subscriptproduct𝑣𝑆12𝜋𝑖subscriptsubscript𝐿𝑣subscript𝑐𝑣superscriptsubscript𝔉𝑣𝑧subscript𝑠𝑣𝑎subscript𝛼𝑣subscript𝑠𝑣differential-dsubscript𝜇𝑣subscript𝑠𝑣𝑓𝑎𝑎superscript𝔞101\displaystyle\prod_{v\in\Sigma_{\rm fin}-S}|{\mathfrak{F}}_{v}^{(z)}(a)|\prod_{v\in\Sigma_{\infty}}|\tilde{\mathfrak{F}}_{v}^{(z)}(a)|\,\prod_{v\in S}\left|\tfrac{1}{2\pi i}\textstyle{\int}_{L_{v}(c_{v})}{\mathfrak{F}}_{v}^{(z)}(s_{v},a)\,\alpha_{v}(s_{v})\,{{d}}\mu_{v}(s_{v})\right|\ll f(a),\quad a\in{\mathfrak{a}}^{-1}-\{0,1\}

uniformly on |Re(z)|σRe𝑧𝜎|\operatorname{Re}(z)|\leqslant\sigma with f(a)=vS(𝔫)Σfv(a(v))𝑓𝑎subscriptproduct𝑣𝑆𝔫subscriptΣsubscript𝑓𝑣superscript𝑎𝑣f(a)=\prod_{v\in S({\mathfrak{n}})\cup\Sigma_{\infty}}f_{v}(a^{(v)}), where fv(a(v))=δ(a(v)>0)(1+|a|v)lv2+σ+12+ϵsubscript𝑓𝑣superscript𝑎𝑣𝛿superscript𝑎𝑣0superscript1subscript𝑎𝑣subscript𝑙𝑣2𝜎12italic-ϵf_{v}(a^{(v)})=\delta(a^{(v)}>0)(1+|a|_{v})^{-\frac{l_{v}}{2}+\frac{\sigma+1}{2}+\epsilon} if vΣ𝑣subscriptΣv\in\Sigma_{\infty} and fv(a(v))=4qv+1(qv1/2+1)δ(a(v)1+𝔭v)subscript𝑓𝑣superscript𝑎𝑣4subscript𝑞𝑣1superscriptsuperscriptsubscript𝑞𝑣121𝛿superscript𝑎𝑣1subscript𝔭𝑣f_{v}(a^{(v)})=\frac{4}{q_{v}+1}(q_{v}^{1/2}+1)^{\delta(a^{(v)}\in 1+{\mathfrak{p}}_{v})} if vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}). The sum a𝔞1{0,1}f(a)subscript𝑎superscript𝔞101𝑓𝑎\sum_{a\in{\mathfrak{a}}^{-1}-\{0,1\}}f(a) is majorized by

𝔠|𝔫a𝔞1{0,1}a1𝔠,a1𝔫𝔠1f(a)𝔠|𝔫vS(𝔫𝔠1)4qv+1vS(𝔠)4(qv1/2+1)qv+1{x𝔠𝔞1{0}f(x+1)}.much-less-thansubscriptconditional𝔠𝔫subscript𝑎superscript𝔞101formulae-sequence𝑎1𝔠𝑎1𝔫superscript𝔠1𝑓𝑎subscriptconditional𝔠𝔫subscriptproduct𝑣𝑆𝔫superscript𝔠14subscript𝑞𝑣1subscriptproduct𝑣𝑆𝔠4superscriptsubscript𝑞𝑣121subscript𝑞𝑣1subscript𝑥𝔠superscript𝔞10subscript𝑓𝑥1\displaystyle\sum_{{\mathfrak{c}}|{\mathfrak{n}}}\sum_{\begin{subarray}{c}a\in{\mathfrak{a}}^{-1}-\{0,1\}\\ a-1\in{\mathfrak{c}},a-1\not\in{\mathfrak{n}}{\mathfrak{c}}^{-1}\end{subarray}}f(a)\ll\sum_{{\mathfrak{c}}|{\mathfrak{n}}}\prod_{v\in S({\mathfrak{n}}{\mathfrak{c}}^{-1})}\tfrac{4}{q_{v}+1}\prod_{v\in S({\mathfrak{c}})}\tfrac{4(q_{v}^{1/2}+1)}{q_{v}+1}\,\biggl{\{}\sum_{x\in{\mathfrak{c}}{\mathfrak{a}}^{-1}-\{0\}}f_{\infty}(x+1)\biggr{\}}.

Let T(𝔫)𝑇𝔫T({\mathfrak{n}}) denote the majorant. Since ι(𝔠𝔞1)ι(𝔞1)subscript𝜄𝔠superscript𝔞1subscript𝜄superscript𝔞1\iota_{\infty}({\mathfrak{c}}{\mathfrak{a}}^{-1})\subset\iota_{\infty}({\mathfrak{a}}^{-1}) are {\mathbb{Z}}-lattices of full rank in ΣsuperscriptsubscriptΣ{\mathbb{R}}^{\Sigma_{\infty}}, we apply Lemma 7.19 to estimate the sum x𝔠𝔞1{0}f(x+1)subscript𝑥𝔠superscript𝔞10subscript𝑓𝑥1\sum_{x\in{\mathfrak{c}}{\mathfrak{a}}^{-1}-\{0\}}f_{\infty}(x+1) by N(𝔞)×N(𝔠𝔞1){1l¯/2+(σ+1)/2+ϵ}/dF{\operatorname{N}}({\mathfrak{a}})\times{\operatorname{N}}({\mathfrak{c}}{\mathfrak{a}}^{-1})^{\{1-{\underline{l}}/2+(\sigma+1)/2+\epsilon\}/d_{F}} uniformly in the ideals 𝔠𝔠{\mathfrak{c}} and 𝔞𝔞{\mathfrak{a}}. Therefore,

T(𝔫)𝑇𝔫\displaystyle T({\mathfrak{n}}) 𝔠|𝔫N(𝔠1𝔫)1+ϵ/2×N(𝔠)1/2+ϵ/2×N(𝔠)(1l¯/2+(σ+1)/2+ϵ)/dF\displaystyle\ll\sum_{{\mathfrak{c}}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{c}}^{-1}{\mathfrak{n}})^{-1+\epsilon/2}\times{\operatorname{N}}({\mathfrak{c}})^{-1/2+\epsilon/2}\times{\operatorname{N}}({\mathfrak{c}})^{(1-{\underline{l}}/2+(\sigma+1)/2+\epsilon)/d_{F}}
N(𝔫)1+ϵ/2𝔠|𝔫N(𝔠)1/2+(1l¯/2+(σ+1)/2+ϵ)/dF.\displaystyle\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon/2}\sum_{{\mathfrak{c}}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{c}})^{1/2+(1-{\underline{l}}/2+(\sigma+1)/2+\epsilon)/d_{F}}.

Suppose l¯>dF+3+σ¯𝑙subscript𝑑𝐹3𝜎{\underline{l}}>d_{F}+3+\sigma. Then we can choose ϵ>0italic-ϵ0\epsilon>0 so that 1/2+(1l¯/2+(σ+1)/2+ϵ)/dF0121¯𝑙2𝜎12italic-ϵsubscript𝑑𝐹01/2+(1-{\underline{l}}/2+(\sigma+1)/2+\epsilon)/d_{F}\leqslant 0. Thus,

T(𝔫)N(𝔫)1+ϵ/2×𝔠|𝔫1N(𝔫)1+ϵ.\displaystyle T({\mathfrak{n}})\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon/2}\times\sum_{{\mathfrak{c}}|{\mathfrak{n}}}1\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}.

Suppose l¯dF+3+σ¯𝑙subscript𝑑𝐹3𝜎{\underline{l}}\leqslant d_{F}+3+\sigma. Then 1/2+(1l¯/2+(σ+1)/2+ϵ)/dF>0121¯𝑙2𝜎12italic-ϵsubscript𝑑𝐹01/2+(1-{\underline{l}}/2+(\sigma+1)/2+\epsilon)/d_{F}>0. Thus,

T(𝔫)𝑇𝔫\displaystyle T({\mathfrak{n}}) N(𝔫)1+ϵ/2×N(𝔫)1/2+(1l¯/2+(σ+1)/2+ϵ)/dF𝔠|𝔫1N(𝔫)1/2+ϵ+(1l¯/2+(σ+1)/2+ϵ)/dF.\displaystyle\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon/2}\times{\operatorname{N}}({\mathfrak{n}})^{1/2+(1-{\underline{l}}/2+(\sigma+1)/2+\epsilon)/d_{F}}\sum_{{\mathfrak{c}}|{\mathfrak{n}}}1\ll{\operatorname{N}}({\mathfrak{n}})^{-1/2+\epsilon+(1-{\underline{l}}/2+(\sigma+1)/2+\epsilon)/d_{F}}.

This completes the proof. ∎

Lemma 8.3.

Set δ=1𝛿1\delta=1 if l¯>dF+2¯𝑙subscript𝑑𝐹2{\underline{l}}>d_{F}+2 and δ=1/2+(l¯2)/(2dF)𝛿12¯𝑙22subscript𝑑𝐹\delta=1/2+({\underline{l}}-2)/(2d_{F}) if dF+2l¯4subscript𝑑𝐹2¯𝑙4d_{F}+2\geqslant{\underline{l}}\geqslant 4. Then 1/2<δ112𝛿11/2<\delta\leqslant 1, and for any small ϵ>0italic-ϵ0\epsilon>0,

|𝕁ell0(𝔫|α,z)|N(𝔫)δ+ϵ+N(𝔫)1+ϵ\displaystyle|{\mathbb{J}}_{\rm ell}^{0}({\mathfrak{n}}|\alpha,z)|\ll{\operatorname{N}}({\mathfrak{n}})^{-\delta+\epsilon}+{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}

uniformly in z𝑧z on the strip |Re(z)|1Re𝑧1|\operatorname{Re}(z)|\leqslant 1.

Proof.

In this proof, we set d=dF𝑑subscript𝑑𝐹d=d_{F}. From (7.15) and Proposition 7.7, by changing the order of the integral and the summation,

(8.5) 𝕁ell0(𝔫|α,z)=superscriptsubscript𝕁ell0conditional𝔫𝛼𝑧absent\displaystyle{\mathbb{J}}_{\rm ell}^{0}({\mathfrak{n}}|\alpha,z)=\, 12γ=(t:n)F𝒬FIrr𝒬FSDFz/2L(z+12,εt24n)N(𝔇t24n)z+14𝐃γ(z){vΣFS𝔈~v(z)(γ^v)}\displaystyle\frac{1}{2}\sum_{\gamma=(t:n)_{F}\in{\mathcal{Q}}_{F}^{\rm Irr}\cap{\mathcal{Q}}_{F}^{S}}D_{F}^{z/2}L\left(\tfrac{z+1}{2},\varepsilon_{t^{2}-4n}\right){\operatorname{N}}({\mathfrak{D}}_{t^{2}-4n})^{\frac{z+1}{4}}{\bf D}_{\gamma}(z)\{\prod_{v\in\Sigma_{F}-S}\tilde{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})\}
(8.6) ×vS12πiL(cv)𝔈~v(z)(sv;γ^v)αv(sv)dμv(sv).\displaystyle\times\prod_{v\in S}\tfrac{1}{2\pi i}\textstyle{\int}_{L(c_{v})}\tilde{\mathfrak{E}}_{v}^{(z)}(s_{v};\hat{\gamma}_{v})\,\alpha_{v}(s_{v})\,{{d}}\mu_{v}(s_{v}).

Here 𝐃γ(z)subscript𝐃𝛾𝑧{\bf D}_{\gamma}(z) is the dyadic factor of the formula of EΔ(z;12)superscript𝐸Δ𝑧subscript12E^{\Delta}(z;1_{2}) in Proposition 7.7, and 𝔈~v(z)(γ^v)superscriptsubscript~𝔈𝑣𝑧subscript^𝛾𝑣\tilde{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) equals |2mv|v1𝔈v(z)(γ^v)superscriptsubscript2subscript𝑚𝑣𝑣1superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣|2m_{v}|_{v}^{-1}{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) or |mv|v1𝔈v(z)(γ^v)superscriptsubscriptsubscript𝑚𝑣𝑣1superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣|m_{v}|_{v}^{-1}{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) according to “Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=1 or vΣ𝑣subscriptΣv\in\Sigma_{\infty}” or not, respectively. We choose the representatives (t,εni,ν)𝑡𝜀subscript𝑛𝑖𝜈(t,\varepsilon n_{i,\nu}) of 𝒬FIrr𝒬FSsuperscriptsubscript𝒬𝐹Irrsuperscriptsubscript𝒬𝐹𝑆{\mathcal{Q}}_{F}^{\rm Irr}\cap{\mathcal{Q}}_{F}^{S} as in Lemma 7.17. For vS𝑣𝑆v\in S such that |t|v2>|ni,ν|vsuperscriptsubscript𝑡𝑣2subscriptsubscript𝑛𝑖𝜈𝑣|t|_{v}^{2}>|n_{i,\nu}|_{v}, we have |t|v=1subscript𝑡𝑣1|t|_{v}=1 from min(2ordv(t),νv){0,1}2subscriptord𝑣𝑡subscript𝜈𝑣01\min(2\operatorname{ord}_{v}(t),\nu_{v})\in\{0,1\}; since v𝑣v is non-dyadic and t24εni,νsuperscript𝑡24𝜀subscript𝑛𝑖𝜈t^{2}-4\varepsilon n_{i,\nu} is a square residue modulo 𝔭vsubscript𝔭𝑣{\mathfrak{p}}_{v}, we have t24εni,ν(Fv×)2superscript𝑡24𝜀subscript𝑛𝑖𝜈superscriptsuperscriptsubscript𝐹𝑣2t^{2}-4\varepsilon n_{i,\nu}\in(F_{v}^{\times})^{2} by Lemma 7.1. Then Proposition 7.12 implies |nmv2|v=|ni,ν|v=qvνvsubscript𝑛superscriptsubscript𝑚𝑣2𝑣subscriptsubscript𝑛𝑖𝜈𝑣superscriptsubscript𝑞𝑣subscript𝜈𝑣|\tfrac{n}{m_{v}^{2}}|_{v}=|n_{i,\nu}|_{v}=q_{v}^{-\nu_{v}}. From Theorem 7.9 and Lemma 8.1,

(8.7) 12πiL(cv)𝔈~v(z)(sv;γ^v)σk(sv)𝑑μv(sv)=012𝜋𝑖subscript𝐿subscript𝑐𝑣superscriptsubscript~𝔈𝑣𝑧subscript𝑠𝑣subscript^𝛾𝑣subscript𝜎𝑘subscript𝑠𝑣differential-dsubscript𝜇𝑣subscript𝑠𝑣0\displaystyle\tfrac{1}{2\pi i}\textstyle{\int}_{L(c_{v})}\tilde{\mathfrak{E}}_{v}^{(z)}(s_{v};\hat{\gamma}_{v})\,\sigma_{k}(s_{v})\,{{d}}\mu_{v}(s_{v})=0

unless |nmv2|vqvksubscript𝑛superscriptsubscript𝑚𝑣2𝑣superscriptsubscript𝑞𝑣𝑘|\frac{n}{m_{v}^{2}}|_{v}\geqslant q_{v}^{-k} for all vS𝑣𝑆v\in S. Thus we may suppose νvnv(=ordv(𝔞))subscript𝜈𝑣annotatedsubscript𝑛𝑣absentsubscriptord𝑣𝔞\nu_{v}\leqslant n_{v}(=\operatorname{ord}_{v}({\mathfrak{a}})) for all vS𝑣𝑆v\in S such that |ni,ν|v|t|v2subscriptsubscript𝑛𝑖𝜈𝑣superscriptsubscript𝑡𝑣2|n_{i,\nu}|_{v}\leqslant|t|_{v}^{2}. Combined with the constraint min(2ordv(t),νv){0,1}2subscriptord𝑣𝑡subscript𝜈𝑣01\min(2\operatorname{ord}_{v}(t),\nu_{v})\in\{0,1\} (vS)𝑣𝑆(v\in S), this implies the vanishing of the S𝑆S-factor (8.6) except for finitely many n=εni,ν𝑛𝜀subscript𝑛𝑖𝜈n=\varepsilon n_{i,\nu}. Thus |𝕁ell0(𝔫|α,z)||{\mathbb{J}}_{\rm ell}^{0}({\mathfrak{n}}|\alpha,z)| is majorized by the sum of all Ξ(z,𝐜)(𝔠,𝔫1,i,εni,ν)superscriptΞ𝑧𝐜𝔠subscript𝔫1𝑖𝜀subscript𝑛𝑖𝜈\Xi^{(z,{\mathbf{c}})}({\mathfrak{c}},{\mathfrak{n}}_{1},i,\varepsilon n_{i,\nu}) with νvordv(𝔞)subscript𝜈𝑣subscriptord𝑣𝔞\nu_{v}\leqslant\operatorname{ord}_{v}({\mathfrak{a}}) for all vS𝑣𝑆v\in S. Then we invoke the estimate Lemma 7.20 for each of these to obtain the majorization |𝕁ell0(𝔫|α,z)|ϵiΣ(𝔫,Ni)+N(𝔫)1+ϵ|{\mathbb{J}}_{\rm ell}^{0}({\mathfrak{n}}|\alpha,z)|\ll_{\epsilon}\sum_{i}\Sigma({\mathfrak{n}},N_{i})+{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon} uniformly in z𝑧z with Re(z)[2l¯+3+6ϵ,l¯1/2ϵ]Re𝑧2¯𝑙36italic-ϵ¯𝑙12italic-ϵ\operatorname{Re}(z)\in[-2\underline{l}+3+6\epsilon,\underline{l}-1/2-\epsilon] and

Σ(𝔫,Ni)Σ𝔫subscript𝑁𝑖\displaystyle\Sigma({\mathfrak{n}},N_{i}) =𝔫1|𝔫𝔠|𝔫1N(𝔫)1+ϵN(𝔫1)1/2+ϵ/2max(max(0,N(𝔫12𝔠2)1/d4Ni1/d),N(𝔠)1/d)1L,\displaystyle=\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}\sum_{{\mathfrak{c}}|{\mathfrak{n}}_{1}}{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}{\operatorname{N}}({\mathfrak{n}}_{1})^{1/2+\epsilon/2}\max\biggl{(}\sqrt{\max(0,{\operatorname{N}}({\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d}-4N_{i}^{1/d})},\,{\operatorname{N}}({\mathfrak{c}})^{1/d}\biggr{)}^{1-L},

where L=L¯(z)=l¯1+ϱ(z)22ϵ𝐿¯𝐿𝑧¯𝑙1italic-ϱ𝑧22italic-ϵL={\underline{L}}(z)={\underline{l}}-\frac{1+\varrho(z)}{2}-2\epsilon and Ni=|N(ni,ν)|subscript𝑁𝑖Nsubscript𝑛𝑖𝜈N_{i}=|{\operatorname{N}}(n_{i,\nu})|. We divide the sum Σ(𝔫,N)Σ𝔫𝑁\Sigma({\mathfrak{n}},N) to two parts ΣIsubscriptΣ𝐼\Sigma_{I} and ΣIIsubscriptΣ𝐼𝐼\Sigma_{II} according as N(𝔫12𝔠2)1/d8N1/d{\operatorname{N}}({\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d}\geqslant 8N^{1/d} or not. Set M=8d/2N1/2𝑀superscript8𝑑2superscript𝑁12M=8^{d/2}N^{1/2}. Then ΣIsubscriptΣ𝐼\Sigma_{I} is over those pairs of ideals (𝔫1,𝔠)subscript𝔫1𝔠({\mathfrak{n}}_{1},{\mathfrak{c}}) such that 𝔫1|𝔫conditionalsubscript𝔫1𝔫{\mathfrak{n}}_{1}|{\mathfrak{n}}, 𝔠|𝔫1conditional𝔠subscript𝔫1{\mathfrak{c}}|{\mathfrak{n}}_{1} and N(𝔫1)MN(𝔠)Nsubscript𝔫1𝑀N𝔠{\operatorname{N}}({\mathfrak{n}}_{1})\geqslant M\,{\operatorname{N}}({\mathfrak{c}}). As such,

max(max(0,N(𝔫12𝔠2)1/d4N1/d),N(𝔠)1/d)\displaystyle\max\biggl{(}\sqrt{\max(0,{\operatorname{N}}({\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d}-4N^{1/d})},\,{\operatorname{N}}({\mathfrak{c}})^{1/d}\biggr{)}
\displaystyle\geqslant max(N(𝔫12𝔠2)1/d12N(𝔫12𝔠2)1/d,N(𝔠)1/d)21/2max(N(𝔫1𝔠1)1/d,N(𝔠)1/d)\displaystyle\max(\sqrt{{\operatorname{N}}({\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d}-\tfrac{1}{2}{\operatorname{N}}({\mathfrak{n}}_{1}^{2}{\mathfrak{c}}^{-2})^{1/d}},\,{\operatorname{N}}({\mathfrak{c}})^{1/d})\geqslant 2^{-1/2}\max({\operatorname{N}}({\mathfrak{n}}_{1}{\mathfrak{c}}^{-1})^{1/d},\,{\operatorname{N}}({\mathfrak{c}})^{1/d})
\displaystyle\geqslant 21/2{N(𝔫1𝔠1)1/d+N(𝔠)1/d}/221/2(N(𝔫1𝔠1)1/d×N(𝔠)1/d)1/2=21/2N(𝔫1)1/(2d).\displaystyle 2^{-1/2}\{{{\operatorname{N}}({\mathfrak{n}}_{1}{\mathfrak{c}}^{-1})^{1/d}+{\operatorname{N}}({\mathfrak{c}})^{1/d}}\}/{2}\geqslant 2^{-1/2}\left({\operatorname{N}}({\mathfrak{n}}_{1}{\mathfrak{c}}^{-1})^{1/d}\times{\operatorname{N}}({\mathfrak{c}})^{1/d}\right)^{1/2}=2^{-1/2}{\operatorname{N}}({\mathfrak{n}}_{1})^{1/(2d)}.

Since 1L1𝐿1-L is non-positive,

ΣIsubscriptΣ𝐼\displaystyle\Sigma_{I} N(𝔫)1+ϵ𝔫1|𝔫N(𝔫1)1/2+ϵ/2𝔠|𝔫1(21/2N(𝔫1)1/(2d))1L\displaystyle\leqslant{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{n}}_{1})^{1/2+\epsilon/2}\sum_{{\mathfrak{c}}|{\mathfrak{n}}_{1}}(2^{-1/2}{\operatorname{N}}({\mathfrak{n}}_{1})^{1/(2d)})^{1-L}
N(𝔫)1+ϵ𝔫1|𝔫𝔠|𝔫1N(𝔫1)1/2+ϵ/2+(1L)/(2d)N(𝔫)1+ϵ𝔫1|𝔫N(𝔫1)ϵ+1/2+(1L)/(2d),\displaystyle\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}\sum_{{\mathfrak{c}}|{\mathfrak{n}}_{1}}{\operatorname{N}}({\mathfrak{n}}_{1})^{1/2+\epsilon/2+(1-L)/(2d)}\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{n}}_{1})^{\epsilon+1/2+(1-L)/(2d)},

where we use the estimate 𝔠|𝔫11N(𝔫1)ϵ/2\sum_{{\mathfrak{c}}|{\mathfrak{n}}_{1}}1\ll{\operatorname{N}}({\mathfrak{n}}_{1})^{\epsilon/2}.

The sum ΣIIsubscriptΣ𝐼𝐼\Sigma_{II} is over all pairs of ideals (𝔫1,𝔠)subscript𝔫1𝔠({\mathfrak{n}}_{1},{\mathfrak{c}}) such that 𝔫1|𝔫conditionalsubscript𝔫1𝔫{\mathfrak{n}}_{1}|{\mathfrak{n}}, 𝔠|𝔫1conditional𝔠subscript𝔫1{\mathfrak{c}}|{\mathfrak{n}}_{1}, M1N(𝔫1)N(𝔠)N(𝔫1)superscript𝑀1Nsubscript𝔫1N𝔠Nsubscript𝔫1M^{-1}\,{\operatorname{N}}({\mathfrak{n}}_{1})\leqslant{\operatorname{N}}({\mathfrak{c}})\leqslant{\operatorname{N}}({\mathfrak{n}}_{1}). By max(A,B)B𝐴𝐵𝐵\max(A,B)\geqslant B and by noting 1L01𝐿01-L\leqslant 0, we have trivially

ΣIIsubscriptΣ𝐼𝐼\displaystyle\Sigma_{II} 𝔫1|𝔫𝔠|𝔫1N(𝔫)1+ϵN(𝔫1)1/2+ϵ/2(N(𝔠)1/d)1L𝔫1|𝔫𝔠|𝔫1N(𝔫)1+ϵN(𝔫1)1/2+ϵ/2+(1L)/d\displaystyle\leqslant\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}\sum_{{\mathfrak{c}}|{\mathfrak{n}}_{1}}{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}{\operatorname{N}}({\mathfrak{n}}_{1})^{1/2+\epsilon/2}({\operatorname{N}}({\mathfrak{c}})^{1/d})^{1-L}\ll\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}\sum_{{\mathfrak{c}}|{\mathfrak{n}}_{1}}{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}{\operatorname{N}}({\mathfrak{n}}_{1})^{1/2+\epsilon/2+(1-L)/d}
N(𝔫)1+ϵ𝔫1|𝔫N(𝔫1)ϵ+1/2+(1L)/dN(𝔫)1+ϵ𝔫1|𝔫N(𝔫1)ϵ+1/2+(1L)/(2d).\displaystyle\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{n}}_{1})^{\epsilon+1/2+(1-L)/d}\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{n}}_{1})^{\epsilon+1/2+(1-L)/(2d)}.

Thus Σ(𝔫,N)=ΣI+ΣIIN(𝔫)1+ϵ𝔫1|𝔫N(𝔫1)ϵ+1/2+(1L)/(2d).\Sigma({\mathfrak{n}},N)=\Sigma_{I}+\Sigma_{II}\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{n}}_{1})^{\epsilon+1/2+(1-L)/(2d)}.

Suppose l¯>d+2¯𝑙𝑑2{\underline{l}}>d+2. Then for sufficiently small ϵ>0italic-ϵ0\epsilon>0, we have ϵ+1/2+(1L)/(2d)0italic-ϵ121𝐿2𝑑0\epsilon+1/2+(1-L)/(2d)\leqslant 0 for Re(z)[0,1]Re𝑧01\operatorname{Re}(z)\in[0,1]. Then

Σ(𝔫,N)ϵN(𝔫)1+ϵ𝔫1|𝔫N(𝔫1)ϵ+1/2+(1L)/(2d)N(𝔫)1+ϵ𝔫1|𝔫1N(𝔫)1+2ϵ.\Sigma({\mathfrak{n}},N)\ll_{\epsilon}{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}{\operatorname{N}}({\mathfrak{n}}_{1})^{\epsilon+1/2+(1-L)/(2d)}\leqslant{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}1\leqslant{\operatorname{N}}({\mathfrak{n}})^{-1+2\epsilon}.

Suppose l¯d+2¯𝑙𝑑2{\underline{l}}\leqslant d+2. Then ϵ+1/2+(1L)/(2d)>0italic-ϵ121𝐿2𝑑0\epsilon+1/2+(1-L)/(2d)>0 for Re(z)[0,1]Re𝑧01\operatorname{Re}(z)\in[0,1]. Thus

Σ(𝔫,N)N(𝔫)1+ϵ×N(𝔫)ϵ+1/2+(1L)/(2d)𝔫1|𝔫1N(𝔫)1/2+3ϵ+(1L)/(2d).\displaystyle\Sigma({\mathfrak{n}},N)\ll{\operatorname{N}}({\mathfrak{n}})^{-1+\epsilon}\times{\operatorname{N}}({\mathfrak{n}})^{\epsilon+1/2+(1-L)/(2d)}\sum_{{\mathfrak{n}}_{1}|{\mathfrak{n}}}1\ll{\operatorname{N}}({\mathfrak{n}})^{-1/2+3\epsilon+(1-L)/(2d)}.

Note L=l¯12ϵ𝐿¯𝑙12italic-ϵL={\underline{l}}-1-2\epsilon for |Re(z)|1Re𝑧1|\operatorname{Re}(z)|\leqslant 1. This completes the proof. ∎

Remark : With a bit more work, it can be shown that the implied constants in Lemmas 8.2 and 8.3 are taken to be of the form α𝒜(𝔞)N(𝔞)N\|\alpha\|_{{\mathcal{A}}({\mathfrak{a}})}{\operatorname{N}}({\mathfrak{a}})^{N} with 𝒜(𝔞)\|\,\|_{{\mathcal{A}}({\mathfrak{a}})} being a norm on the finite dimensional space 𝒜(𝔞)𝒜𝔞{\mathcal{A}}({\mathfrak{a}}).

Theorem 1.3 (1) for f𝒫(ΩS)𝑓𝒫subscriptΩ𝑆f\in{\mathcal{P}}(\Omega_{S}) follows from Corollary 1.2 combined with Lemmas 8.2 and 8.3 immediately. The assertion of Theorem 1.3 (1) for fC(ΩS)𝑓𝐶subscriptΩ𝑆f\in C(\Omega_{S}) is proved by a standard argement (cf. [19, Proposition 2]), where the non-negativity of the measure Λl,𝔫(z)superscriptsubscriptΛ𝑙𝔫𝑧\Lambda_{l,{\mathfrak{n}}}^{(z)} is indispensable. Indeed, for any fC(ΩS)𝑓𝐶subscriptΩ𝑆f\in C(\Omega_{S}) and any ϵ>0italic-ϵ0\epsilon>0, the Weierstrass approximation theorem allows us to take a polynomial function Pϵ𝒫(ΩS)subscript𝑃italic-ϵ𝒫subscriptΩ𝑆P_{\epsilon}\in{\mathcal{P}}(\Omega_{S}) such that supxΩS|f(x)Pϵ(x)|<ϵsubscriptsupremum𝑥subscriptΩ𝑆𝑓𝑥subscript𝑃italic-ϵ𝑥italic-ϵ\sup_{x\in\Omega_{S}}|f(x)-P_{\epsilon}(x)|<\epsilon. We set Λ𝔫(z)=r(z)1(Cl(z))1Λl,𝔫(z)superscriptsubscriptΛ𝔫𝑧𝑟superscript𝑧1superscriptsuperscriptsubscript𝐶𝑙𝑧1superscriptsubscriptΛ𝑙𝔫𝑧\Lambda_{\mathfrak{n}}^{(z)}=r(z)^{-1}(C_{l}^{(z)})^{-1}\Lambda_{l,{\mathfrak{n}}}^{(z)} and λS(z)=vSλv(z)\lambda_{S}^{(z)}=\otimes_{v\in S}\lambda_{v}^{(z)} for simplicity. By the condition (P), we have |Λ𝔫(z),fPϵ|Λ𝔫(z),1sup|fPϵ|<Λ𝔫(z),1ϵsuperscriptsubscriptΛ𝔫𝑧𝑓subscript𝑃italic-ϵsuperscriptsubscriptΛ𝔫𝑧1supremum𝑓subscript𝑃italic-ϵsuperscriptsubscriptΛ𝔫𝑧1italic-ϵ|\langle\Lambda_{{\mathfrak{n}}}^{(z)},f-P_{\epsilon}\rangle|\leqslant\langle\Lambda_{{\mathfrak{n}}}^{(z)},1\rangle\sup|f-P_{\epsilon}|<\langle\Lambda_{{\mathfrak{n}}}^{(z)},1\rangle\epsilon. Furthermore, there exists Nϵ>0subscript𝑁italic-ϵ0N_{\epsilon}>0 such that supz|Λ𝔫(z)λS(z),Pϵ|<ϵsubscriptsupremum𝑧superscriptsubscriptΛ𝔫𝑧superscriptsubscript𝜆𝑆𝑧subscript𝑃italic-ϵitalic-ϵ\sup_{z}|\langle\Lambda_{{\mathfrak{n}}}^{(z)}-\lambda_{S}^{(z)},P_{\epsilon}\rangle|<\epsilon and supz|Λ𝔫(z)λS(z),1|<λS(z),1subscriptsupremum𝑧superscriptsubscriptΛ𝔫𝑧superscriptsubscript𝜆𝑆𝑧1superscriptsubscript𝜆𝑆𝑧1\sup_{z}|\langle\Lambda_{\mathfrak{n}}^{(z)}-\lambda_{S}^{(z)},1\rangle|<\langle\lambda_{S}^{(z)},1\rangle for all 𝔫𝔫{\mathfrak{n}} with N(𝔫)>NϵN𝔫subscript𝑁italic-ϵ{\operatorname{N}}({\mathfrak{n}})>N_{\epsilon}. As a result, we have

supz|Λ𝔫(z)λS(z),f|subscriptsupremum𝑧superscriptsubscriptΛ𝔫𝑧superscriptsubscript𝜆𝑆𝑧𝑓absent\displaystyle\sup_{z}|\langle\Lambda_{{\mathfrak{n}}}^{(z)}-\lambda_{S}^{(z)},f\rangle|\leqslant supz|Λ𝔫(z),fPϵ|+supz|Λ𝔫(z)λS(z),Pϵ|+supz|λS(z),Pϵf|subscriptsupremum𝑧superscriptsubscriptΛ𝔫𝑧𝑓subscript𝑃italic-ϵsubscriptsupremum𝑧superscriptsubscriptΛ𝔫𝑧superscriptsubscript𝜆𝑆𝑧subscript𝑃italic-ϵsubscriptsupremum𝑧superscriptsubscript𝜆𝑆𝑧subscript𝑃italic-ϵ𝑓\displaystyle\sup_{z}|\langle\Lambda_{{\mathfrak{n}}}^{(z)},f-P_{\epsilon}\rangle|+\sup_{z}|\langle\Lambda_{{\mathfrak{n}}}^{(z)}-\lambda_{S}^{(z)},P_{\epsilon}\rangle|+\sup_{z}|\langle\lambda_{S}^{(z)},P_{\epsilon}-f\rangle|
<\displaystyle< (supz3λS(z),1+1)ϵ.subscriptsupremum𝑧3superscriptsubscript𝜆𝑆𝑧11italic-ϵ\displaystyle(\sup_{z}3\langle\lambda_{S}^{(z)},1\rangle+1)\epsilon.

This completes the proof as z𝑧z varies in the compact set [0,min(1,σ)]01𝜎[0,\min(1,\sigma)].

As for the proof of Theorem 1.3 (2), we first show that, under the assumption (P), the limit formula in Theorem 1.3 (1) is valid even for f=chJ𝑓subscriptch𝐽f={\rm ch}_{J} with J=vS[tv,tv]𝐽subscriptproduct𝑣𝑆subscript𝑡𝑣subscriptsuperscript𝑡𝑣J=\prod_{v\in S}[t_{v},t^{\prime}_{v}], which is discontinuous. We set Λ𝔫(z)=r(z)1(Cl(z))1Λl,𝔫(z)superscriptsubscriptΛ𝔫𝑧𝑟superscript𝑧1superscriptsuperscriptsubscript𝐶𝑙𝑧1superscriptsubscriptΛ𝑙𝔫𝑧\Lambda_{\mathfrak{n}}^{(z)}=r(z)^{-1}(C_{l}^{(z)})^{-1}\Lambda_{l,{\mathfrak{n}}}^{(z)} and λS(z)=vSλv(z)\lambda_{S}^{(z)}=\otimes_{v\in S}\lambda_{v}^{(z)} for simplicity. Put Jδ=ΩSvS[tvδ,tv+δ]subscript𝐽𝛿subscriptΩ𝑆subscriptproduct𝑣𝑆subscript𝑡𝑣𝛿subscriptsuperscript𝑡𝑣𝛿J_{\delta}=\Omega_{S}\cap\prod_{v\in S}[t_{v}-\delta,t^{\prime}_{v}+\delta] for any δ𝛿\delta\in\mathbb{R}. Then [0,1]zvol(JδJδ;λS(z))contains01𝑧maps-tovolsubscript𝐽𝛿subscript𝐽𝛿superscriptsubscript𝜆𝑆𝑧[0,1]\ni z\mapsto{\operatorname{vol}}(J_{\delta}-J_{-\delta};\lambda_{S}^{(z)}) is monotonously increasing for each δ>0𝛿0\delta>0. For any fixed ϵ>0italic-ϵ0\epsilon>0, take δ>0𝛿0\delta>0 such that vol(JδJδ;λS(1))<ϵvolsubscript𝐽𝛿subscript𝐽𝛿superscriptsubscript𝜆𝑆1italic-ϵ{\operatorname{vol}}(J_{\delta}-J_{-\delta};\lambda_{S}^{(1)})<\epsilon, and fϵ,gϵC(ΩS)subscript𝑓italic-ϵsubscript𝑔italic-ϵ𝐶subscriptΩ𝑆f_{\epsilon},g_{\epsilon}\in C(\Omega_{S}) such that chJδgϵchJfϵchJδsubscriptchsubscript𝐽𝛿subscript𝑔italic-ϵsubscriptch𝐽subscript𝑓italic-ϵsubscriptchsubscript𝐽𝛿{\rm ch}_{J_{-\delta}}\leqslant g_{\epsilon}\leqslant{\rm ch}_{J}\leqslant f_{\epsilon}\leqslant{\rm ch}_{J_{\delta}}. By Theorem 1.3 (1), there exists Nϵ>0subscript𝑁italic-ϵ0N_{\epsilon}>0 such that both |Λ𝔫(z)λS(z),fϵ|<ϵsuperscriptsubscriptΛ𝔫𝑧superscriptsubscript𝜆𝑆𝑧subscript𝑓italic-ϵitalic-ϵ|\langle\Lambda_{{\mathfrak{n}}}^{(z)}-\lambda_{S}^{(z)},f_{\epsilon}\rangle|<\epsilon and Λ𝔫(z),fϵgϵ<λS(z),fϵgϵ+ϵsuperscriptsubscriptΛ𝔫𝑧subscript𝑓italic-ϵsubscript𝑔italic-ϵsuperscriptsubscript𝜆𝑆𝑧subscript𝑓italic-ϵsubscript𝑔italic-ϵitalic-ϵ\langle\Lambda_{{\mathfrak{n}}}^{(z)},f_{\epsilon}-g_{\epsilon}\rangle<\langle\lambda_{S}^{(z)},f_{\epsilon}-g_{\epsilon}\rangle+\epsilon hold for all 𝔫𝔫{\mathfrak{n}} with N(𝔫)>NϵN𝔫subscript𝑁italic-ϵ{\operatorname{N}}({\mathfrak{n}})>N_{\epsilon} and for all z[0,min(1,σ)]𝑧01𝜎z\in[0,\min(1,\sigma)]. Then, |Λ𝔫(z)λS(z),chJ|superscriptsubscriptΛ𝔫𝑧superscriptsubscript𝜆𝑆𝑧subscriptch𝐽|\langle\Lambda_{{\mathfrak{n}}}^{(z)}-\lambda_{S}^{(z)},{\rm ch}_{J}\rangle| for N(𝔫)>NϵN𝔫subscript𝑁italic-ϵ{\operatorname{N}}({\mathfrak{n}})>N_{\epsilon} with z[0,min(1,σ)]𝑧01𝜎z\in[0,\min(1,\sigma)] is majorized by

|Λ𝔫(z),chJfϵ|+|Λ𝔫(z)λS(z),fϵ|+|λS(z),fϵchJ|Λ𝔫(z),fϵgϵ+ϵ+vol(JδJ;λS(1))superscriptsubscriptΛ𝔫𝑧subscriptch𝐽subscript𝑓italic-ϵsuperscriptsubscriptΛ𝔫𝑧superscriptsubscript𝜆𝑆𝑧subscript𝑓italic-ϵsuperscriptsubscript𝜆𝑆𝑧subscript𝑓italic-ϵsubscriptch𝐽superscriptsubscriptΛ𝔫𝑧subscript𝑓italic-ϵsubscript𝑔italic-ϵitalic-ϵvolsubscript𝐽𝛿𝐽superscriptsubscript𝜆𝑆1\displaystyle|\langle\Lambda_{{\mathfrak{n}}}^{(z)},{\rm ch}_{J}-f_{\epsilon}\rangle|+|\langle\Lambda_{{\mathfrak{n}}}^{(z)}-\lambda_{S}^{(z)},f_{\epsilon}\rangle|+|\langle\lambda_{S}^{(z)},f_{\epsilon}-{\rm ch}_{J}\rangle|\leqslant\langle\Lambda_{{\mathfrak{n}}}^{(z)},f_{\epsilon}-g_{\epsilon}\rangle+\epsilon+{\operatorname{vol}}(J_{\delta}-J;\lambda_{S}^{(1)})
\displaystyle\leqslant (λS(z),fϵgϵ+ϵ)+2ϵvol(JδJδ;λS(1))+3ϵ<4ϵ.superscriptsubscript𝜆𝑆𝑧subscript𝑓italic-ϵsubscript𝑔italic-ϵitalic-ϵ2italic-ϵvolsubscript𝐽𝛿subscript𝐽𝛿superscriptsubscript𝜆𝑆13italic-ϵ4italic-ϵ\displaystyle(\langle\lambda_{S}^{(z)},f_{\epsilon}-g_{\epsilon}\rangle+\epsilon)+2\epsilon\leqslant{\operatorname{vol}}(J_{\delta}-J_{-\delta};\lambda_{S}^{(1)})+3\epsilon<4\epsilon.

This completes the required convergence for f=chJ𝑓subscriptch𝐽f={\rm ch}_{J}.

Let us return to the proof of Theorem 1.3 (2). Let {\mathcal{I}} be the set of all the prime ideals 𝔫𝔬𝔫𝔬{\mathfrak{n}}\subset\mathfrak{o} relatively prime to 2𝔭Sj=1h𝔞j2subscript𝔭𝑆superscriptsubscriptproduct𝑗1subscript𝔞𝑗2{\mathfrak{p}}_{S}\prod_{j=1}^{h}{\mathfrak{a}}_{j}. Set f=chJ𝑓subscriptch𝐽f={\rm ch}_{J}. As shown above, limN(𝔫)Λl,𝔫(z)(f)>0subscriptN𝔫superscriptsubscriptΛ𝑙𝔫𝑧𝑓0\lim_{{\operatorname{N}}({\mathfrak{n}})\rightarrow\infty}\Lambda_{l,{\mathfrak{n}}}^{(z)}(f)>0 holds with the convergence being uniform in z[0,min(1,σ)]𝑧01𝜎z\in[0,\min(1,\sigma)]. Let us define Λl,𝔫(z),(f)superscriptsubscriptΛ𝑙𝔫𝑧𝑓\Lambda_{l,{\mathfrak{n}}}^{(z),*}(f) by the same formula as Λl,𝔫(z)(f)superscriptsubscriptΛ𝑙𝔫𝑧𝑓\Lambda_{l,{\mathfrak{n}}}^{(z)}(f) reducing the summation range from Πcus(l,𝔫)subscriptΠcus𝑙𝔫\Pi_{\rm cus}(l,{\mathfrak{n}}) to Πcus(l,𝔫)Πcus(l,𝔬)subscriptΠcus𝑙𝔫subscriptΠcus𝑙𝔬\Pi_{\rm cus}(l,{\mathfrak{n}})-\Pi_{\rm cus}(l,{\mathfrak{o}}). By the uniform bound 0<W𝔫(z)(π)ϵN(𝔫)1/2+ϵ0<W_{{\mathfrak{n}}}^{(z)}(\pi)\ll_{\epsilon}{\operatorname{N}}({\mathfrak{n}})^{1/2+\epsilon} (z[0,1],πΠcus(l,𝔬)(z\in[0,1],\,\pi\in\Pi_{\rm cus}(l,\mathfrak{o})), we easiy confirm the difference |Λl,𝔫(z)(f)Λl,𝔫(z)(f)|superscriptsubscriptΛ𝑙𝔫𝑧𝑓superscriptsubscriptΛ𝑙𝔫𝑧𝑓|\Lambda_{l,{\mathfrak{n}}}^{(z)}(f)-\Lambda_{l,{\mathfrak{n}}}^{(z)*}(f)| tends to 00 as N(𝔫)N𝔫{\operatorname{N}}({\mathfrak{n}})\rightarrow\infty uniformly in z[0,1]𝑧01z\in[0,1]. Hence there exists M>0𝑀0M>0 such that Λl,𝔫(z)(f)>0superscriptsubscriptΛ𝑙𝔫𝑧𝑓0\Lambda_{l,{\mathfrak{n}}}^{(z)*}(f)>0 for all z[0,min(1,σ)]𝑧01𝜎z\in[0,\min(1,\sigma)] and for all 𝔫𝔫{\mathfrak{n}}\in{\mathcal{I}} with N(𝔫)>MN𝔫𝑀{\operatorname{N}}({\mathfrak{n}})>M. Hence for any 𝔫𝔫{\mathfrak{n}}\in{\mathcal{I}} with N(𝔫)>MN𝔫𝑀{\operatorname{N}}({\mathfrak{n}})>M and any z[0,min(1,σ)]𝑧01𝜎z\in[0,\min(1,\sigma)], there exists πΠcus(l,𝔫)Πcus(l,𝔬)𝜋subscriptΠcus𝑙𝔫subscriptΠcus𝑙𝔬\pi\in\Pi_{\rm cus}(l,{\mathfrak{n}})-\Pi_{\rm cus}(l,\mathfrak{o}) such that L(z+12,π;Ad)chJ(𝐱S(π))0𝐿𝑧12𝜋Adsubscriptch𝐽subscript𝐱𝑆𝜋0L(\tfrac{z+1}{2},\pi;{\operatorname{Ad}}){\rm ch}_{J}({\mathbf{x}}_{S}(\pi))\neq 0. We have 𝔣π=𝔫subscript𝔣𝜋𝔫{\mathfrak{f}}_{\pi}={\mathfrak{n}} since 𝔫𝔫{\mathfrak{n}} is prime. This completes the proof. We remark that in the proof above the condition (P) is not necessary when J=ΩS𝐽subscriptΩ𝑆J=\Omega_{S}, in which case f𝑓f belongs to 𝒫(ΩS)𝒫subscriptΩ𝑆{\mathcal{P}}(\Omega_{S}).

9. The cuspidal case

Let 𝚽l(𝔫|𝐬,g,h)superscript𝚽𝑙conditional𝔫𝐬𝑔{\mathbf{\Phi}}^{l}({\mathfrak{n}}|{\mathbf{s}},g,h) be the automorphic kernel function constructed in § 2.4. Let ξvξv\xi\cong\otimes_{v}\xi_{v} be an irreducible cuspidal automorphic representation of G𝔸subscript𝐺𝔸G_{\mathbb{A}} with trivial central character which is everywhere unramified, i.e., ξv𝕂v{0}superscriptsubscript𝜉𝑣subscript𝕂𝑣0\xi_{v}^{{\mathbb{K}}_{v}}\not=\{0\} for all vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}. Thus we have a set of numbers νv(ξ)i0(0,1)subscript𝜈𝑣𝜉𝑖subscriptabsent001\nu_{v}(\xi)\in i\mathbb{R}_{\geqslant 0}\cup(0,1) (vΣ)𝑣subscriptΣ(v\in\Sigma_{\infty}) and νv(ξ){iy| 0y2π(logqv)1}{x+iy| 0<x<1,y{0,2π(logqv)1}}subscript𝜈𝑣𝜉conditional-set𝑖𝑦 0𝑦2𝜋superscriptsubscript𝑞𝑣1conditional-set𝑥𝑖𝑦formulae-sequence 0𝑥1𝑦02𝜋superscriptsubscript𝑞𝑣1\nu_{v}(\xi)\in\{iy\ |\ 0\leqslant y\leqslant 2\pi(\log q_{v})^{-1}\}\cup\{x+iy\ |\ 0<x<1,\ y\in\{0,2\pi(\log q_{v})^{-1}\}\} (vΣfin)𝑣subscriptΣfin(v\in\Sigma_{\rm fin}) such that ξvI(||vνv(ξ)/2)\xi_{v}\cong I(|\,|_{v}^{\nu_{v}(\xi)/2}) for all vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}. Let φξ0=φξnewsuperscriptsubscript𝜑𝜉0superscriptsubscript𝜑𝜉new\varphi_{\xi}^{0}=\varphi_{\xi}^{\rm new} be the new vector of ξ𝜉\xi. Since φξ0superscriptsubscript𝜑𝜉0\varphi_{\xi}^{0} is rapidly decreasing on the Siegel set 𝔖1superscript𝔖1{\mathfrak{S}}^{1} of G𝔸1superscriptsubscript𝐺𝔸1G_{\mathbb{A}}^{1}, the integral

(9.1) Z𝔸GF\G𝔸φξ0(g)𝚽l(𝔫|𝐬,g,g)𝑑gsubscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸superscriptsubscript𝜑𝜉0𝑔superscript𝚽𝑙conditional𝔫𝐬𝑔𝑔differential-d𝑔\displaystyle\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}\varphi_{\xi}^{0}(g){\mathbf{\Phi}}^{l}({\mathfrak{n}}|{\mathbf{s}},g,g)\,{{d}}g

converges absolutely. By replacing β(g)subscript𝛽𝑔{\mathcal{E}}_{\beta}(g) with φξ0superscriptsubscript𝜑𝜉0\varphi_{\xi}^{0} in every occurrence, almost all proofs to compute (3.5) so far also work for the integral (9.1) by a slight modification. We end up with a formula resemble to the one in Corollary 1.2. To describe it, we need further notation. For πΠcus(l,𝔫)𝜋subscriptΠcus𝑙𝔫\pi\in\Pi_{\rm cus}(l,{\mathfrak{n}}), we consider the sum of triple product of cusp forms

φξ0(l,𝔫|π)=φπ(l,𝔫)φξ0|φφ¯L2subscriptsuperscriptsubscript𝜑𝜉0𝑙conditional𝔫𝜋subscript𝜑subscript𝜋𝑙𝔫subscriptinner-productsuperscriptsubscript𝜑𝜉0𝜑¯𝜑superscript𝐿2\mathbb{P}_{\varphi_{\xi}^{0}}(l,{\mathfrak{n}}|\pi)=\sum_{\varphi\in{\mathcal{B}}_{\pi}(l,{\mathfrak{n}})}\langle\varphi_{\xi}^{0}|\varphi\,\bar{\varphi}\rangle_{L^{2}}

as (3.7). We modify the definition of 𝐁𝔫(z)(𝐬|Δ;𝔞)subscriptsuperscript𝐁𝑧𝔫conditional𝐬Δ𝔞{\bf B}^{(z)}_{{\mathfrak{n}}}({\mathbf{s}}|\Delta;{\mathfrak{a}}) in § 1.2 by replacing the parameter z𝑧z with νv(ξ)subscript𝜈𝑣𝜉\nu_{v}(\xi) in the v𝑣v-factor for all vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}. i.e.,

𝐁𝔫ξ(𝐬|Δ;𝔞)=vΣfin(SS(𝔫))𝒪0,vΔ,(νv(ξ))(av)vS(𝔫)𝒪1,vΔ,(νv(ξ))(av)vS𝒮vΔ,(νv(ξ))(sv,av).superscriptsubscript𝐁𝔫𝜉conditional𝐬Δ𝔞subscriptproduct𝑣subscriptΣfin𝑆𝑆𝔫superscriptsubscript𝒪0𝑣Δsubscript𝜈𝑣𝜉subscript𝑎𝑣subscriptproduct𝑣𝑆𝔫superscriptsubscript𝒪1𝑣Δsubscript𝜈𝑣𝜉subscript𝑎𝑣subscriptproduct𝑣𝑆superscriptsubscript𝒮𝑣Δsubscript𝜈𝑣𝜉subscript𝑠𝑣subscript𝑎𝑣{\bf B}_{{\mathfrak{n}}}^{\xi}({\mathbf{s}}|\Delta;{\mathfrak{a}})=\prod_{v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}}))}{\mathcal{O}}_{0,v}^{\Delta,(\nu_{v}(\xi))}(a_{v})\prod_{v\in S({\mathfrak{n}})}{\mathcal{O}}_{1,v}^{\Delta,(\nu_{v}(\xi))}(a_{v})\prod_{v\in S}{\mathcal{S}}_{v}^{\Delta,(\nu_{v}(\xi))}(s_{v},a_{v}).

For any quadratic field extension E=F(Δ)𝐸𝐹ΔE=F(\sqrt{\Delta}) with a prescribed square root of ΔF×Δsuperscript𝐹\Delta\in F^{\times}, we define

𝒫Δ(φξ0)=DF1/2N(𝔡E/F)1/2{vΣdyadicΔv0=52(1+νv(ξ))/23(1+2νv(ξ))1}𝔸×E×\𝔸E×φξ0(ιΔ(τ)RΔ1)d×τ.\displaystyle{\mathcal{P}}_{\Delta}(\varphi_{\xi}^{0})=D_{F}^{-1/2}{\operatorname{N}}({\mathfrak{d}}_{E/F})^{1/2}\{\prod_{\begin{subarray}{c}v\in\Sigma_{\rm dyadic}\\ \Delta_{v}^{0}=5\end{subarray}}2^{-(1+\nu_{v}(\xi))/2}3(1+2^{-\nu_{v}(\xi)})^{-1}\}\int_{{\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times}}\varphi_{\xi}^{0}\bigl{(}\iota_{\Delta}(\tau)R_{\Delta}^{-1}\bigr{)}\,{{d}}^{\times}\tau.
Theorem 9.1.

Retain all the assumptions and notation in Theorem 1.1. When minvSRe(sv)>2minvΣlv1subscript𝑣𝑆Resubscript𝑠𝑣2subscript𝑣subscriptΣsubscript𝑙𝑣1\min_{v\in S}\operatorname{Re}(s_{v})>2\min_{v\in\Sigma_{\infty}}l_{v}-1, we have the identity

(1)#SC(l,𝔫)πΠcus(l,𝔫)φξ0(l,𝔫|π)vS{(qv(1+νv(π))/2+qv(1νv(π))/2)(qv(1+sv)/2+qv(1sv)/2)}superscript1#𝑆𝐶𝑙𝔫subscript𝜋subscriptΠcus𝑙𝔫subscriptsuperscriptsubscript𝜑𝜉0𝑙conditional𝔫𝜋subscriptproduct𝑣𝑆superscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜋2superscriptsubscript𝑞𝑣1subscript𝜈𝑣𝜋2superscriptsubscript𝑞𝑣1subscript𝑠𝑣2superscriptsubscript𝑞𝑣1subscript𝑠𝑣2\displaystyle(-1)^{\#S}C(l,{\mathfrak{n}})\,\sum_{\pi\in\Pi_{\rm cus}(l,{\mathfrak{n}})}\frac{\mathbb{P}_{\varphi_{\xi}^{0}}(l,{\mathfrak{n}}|\pi)}{\prod_{v\in S}\{(q_{v}^{(1+\nu_{v}(\pi))/2}+q_{v}^{(1-\nu_{v}(\pi))/2})-(q_{v}^{(1+s_{v})/2}+q_{v}^{(1-s_{v})/2})\}}
=\displaystyle= 𝕁hyp(𝐬,φξ0)+𝕁ell(𝐬,φξ0).subscript𝕁hyp𝐬superscriptsubscript𝜑𝜉0subscript𝕁ell𝐬superscriptsubscript𝜑𝜉0\displaystyle{\mathbb{J}}_{\rm hyp}({\mathbf{s}},\varphi_{\xi}^{0})+{\mathbb{J}}_{\rm ell}({\mathbf{s}},\varphi_{\xi}^{0}).

The right-hand side is given by the absolutely convergent sums

𝕁hyp(𝐬,φξ0)subscript𝕁hyp𝐬superscriptsubscript𝜑𝜉0\displaystyle{\mathbb{J}}_{\rm hyp}({\mathbf{s}},\varphi_{\xi}^{0}) =12DF1L(12,ξ)a𝔬(S)×{1}𝐁𝔫ξ(𝐬|1;a(a1)2𝔬)vΣ𝒪v+,(νv(ξ))((a+1)/(a1)),absent12superscriptsubscript𝐷𝐹1𝐿12𝜉subscript𝑎𝔬superscript𝑆1superscriptsubscript𝐁𝔫𝜉conditional𝐬1𝑎superscript𝑎12𝔬subscriptproduct𝑣subscriptΣsuperscriptsubscript𝒪𝑣subscript𝜈𝑣𝜉𝑎1𝑎1\displaystyle=\tfrac{1}{2}D_{F}^{-1}L\left(\tfrac{1}{2},\xi\right)\sum_{a\in\mathfrak{o}(S)^{\times}-\{1\}}{\bf B}_{{\mathfrak{n}}}^{\xi}({\mathbf{s}}|1;a(a-1)^{-2}\mathfrak{o})\prod_{v\in\Sigma_{\infty}}{\mathcal{O}}_{v}^{+,(\nu_{v}(\xi))}((a+1)/(a-1)),
𝕁ell(𝐬,φξ0)subscript𝕁ell𝐬superscriptsubscript𝜑𝜉0\displaystyle{\mathbb{J}}_{\rm ell}({\mathbf{s}},\varphi_{\xi}^{0}) =12(t:n)F𝒫Δ(φξ0)𝐁𝔫ξ(𝐬|Δ;n𝔣Δ2)vΣ𝒪vsgn(Δ(v)),(νv(ξ))(t|Δ|v1/2),\displaystyle=\tfrac{1}{2}\sum_{(t:n)_{F}}{\mathcal{P}}_{\Delta}(\varphi_{\xi}^{0})\,{\bf B}_{{\mathfrak{n}}}^{\xi}({\mathbf{s}}|\Delta;n{\mathfrak{f}}_{\Delta}^{-2})\,\prod_{v\in\Sigma_{\infty}}{\mathcal{O}}_{v}^{{\operatorname{sgn}}(\Delta^{(v)}),(\nu_{v}(\xi))}(t|\Delta|_{v}^{-1/2}),

where Δ=t24nΔsuperscript𝑡24𝑛\Delta=t^{2}-4n and (t:n)F(t:n)_{F} ranges over the same set as in Corollary 1.2.

Proof.

In accordance with (4.1), the integral (9.1) breaks up to the sum of four terms 𝕁(𝐬,φξ0)=Z𝔸GF\G𝔸φξ0(g)J(𝐬;g)𝑑gsubscript𝕁𝐬superscriptsubscript𝜑𝜉0subscript\subscript𝑍𝔸subscript𝐺𝐹subscript𝐺𝔸superscriptsubscript𝜑𝜉0𝑔subscript𝐽𝐬𝑔differential-d𝑔{\mathbb{J}}_{\natural}({\mathbf{s}},\varphi_{\xi}^{0})=\int_{Z_{\mathbb{A}}G_{F}\backslash G_{\mathbb{A}}}\varphi_{\xi}^{0}(g)J_{\natural}({\mathbf{s}};g)\,{{d}}g with {id,unip,hyp,ell}iduniphypell\natural\in\{{\rm id},{\rm unip},{\rm hyp},{\rm ell}\}. By the cuspidality of φξ0superscriptsubscript𝜑𝜉0\varphi_{\xi}^{0}, it is seen easily that 𝕁id(𝐬;φξ0)=𝕁unip(𝐬;φξ0)=0subscript𝕁id𝐬superscriptsubscript𝜑𝜉0subscript𝕁unip𝐬superscriptsubscript𝜑𝜉00{\mathbb{J}}_{\rm id}({\mathbf{s}};\varphi_{\xi}^{0})={\mathbb{J}}_{\rm unip}({\mathbf{s}};\varphi_{\xi}^{0})=0. By the same argument as in § 6, the hyperbolic term becomes

𝕁hyp(𝐬,φξ0)=12𝒫1(φξ0)aF×{1}vΣF𝔉v(νv(ξ))(a),subscript𝕁hyp𝐬superscriptsubscript𝜑𝜉012subscript𝒫1superscriptsubscript𝜑𝜉0subscript𝑎superscript𝐹1subscriptproduct𝑣subscriptΣ𝐹superscriptsubscript𝔉𝑣subscript𝜈𝑣𝜉𝑎{\mathbb{J}}_{\rm hyp}({\mathbf{s}},\varphi_{\xi}^{0})=\tfrac{1}{2}{\mathcal{P}}_{1}(\varphi_{\xi}^{0})\sum_{a\in F^{\times}-\{1\}}\prod_{v\in\Sigma_{F}}{\mathfrak{F}}_{v}^{(\nu_{v}(\xi))}(a),

with 𝒫1(φξ0)=F×\𝔸×φξ0([t001])d×tsubscript𝒫1superscriptsubscript𝜑𝜉0subscript\superscript𝐹superscript𝔸superscriptsubscript𝜑𝜉0delimited-[]𝑡001superscript𝑑𝑡{\mathcal{P}}_{1}(\varphi_{\xi}^{0})=\int_{F^{\times}\backslash{\mathbb{A}}^{\times}}\varphi_{\xi}^{0}\left(\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}^{\times}t being the Hecke’s integral, which is identified with the central value of L𝐿L-function: 𝒫1(φξ0)=DF1/2L(1/2,ξ)subscript𝒫1superscriptsubscript𝜑𝜉0superscriptsubscript𝐷𝐹12𝐿12𝜉{\mathcal{P}}_{1}(\varphi_{\xi}^{0})=D_{F}^{-1/2}L(1/2,\xi). Since φξ0superscriptsubscript𝜑𝜉0\varphi_{\xi}^{0} has no constant term in the Fourier expansion, the first part of § 6.2 is irrelevant; the absolute convergence is shown as in § 6.4. In the same way as in § 7, the elliptic term becomes

𝕁ell(𝐬,φξ0)=12(t:n)F𝒬FIrr𝒬FS{𝔸×E×\𝔸E×φξ0(ιΔ(τ)RΔ1)d×τ}vΣF𝔈v(νv(ξ))(γ^v).{\mathbb{J}}_{\rm ell}({\mathbf{s}},\varphi_{\xi}^{0})=\tfrac{1}{2}\sum_{(t:n)_{F}\in{\mathcal{Q}}_{F}^{\rm Irr}\cap{\mathcal{Q}}_{F}^{S}}\{\textstyle{\int}_{{\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times}}\varphi_{\xi}^{0}\bigl{(}\iota_{\Delta}(\tau)R_{\Delta}^{-1}\bigr{)}\,{{d}}^{\times}\tau\}\prod_{v\in\Sigma_{F}}{\mathfrak{E}}_{v}^{(\nu_{v}(\xi))}(\hat{\gamma}_{v}).

Since limz1(z1)EΔ(z;12)=DF1/2ζF(2)1Resz=1ζF(z)vol(𝔸×E×\𝔸E×;d×τ)subscript𝑧1𝑧1superscript𝐸Δ𝑧subscript12superscriptsubscript𝐷𝐹12subscript𝜁𝐹superscript21subscriptRes𝑧1subscript𝜁𝐹𝑧vol\superscript𝔸superscript𝐸superscriptsubscript𝔸𝐸superscript𝑑𝜏\lim_{z\rightarrow 1}(z-1)E^{\Delta}(z;1_{2})=D_{F}^{-1/2}\zeta_{F}(2)^{-1}{\rm Res}_{z=1}\zeta_{F}(z)\,{\operatorname{vol}}({\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times};{{d}}^{\times}\tau) (cf. [30, Lemma 2.13]), from Lemma 7.8, we obtain the majorization |𝔸×E×\𝔸E×φξ0(ιΔ(τ)RΔ1)d×τ|vol(𝔸×E×\𝔸E×;d×τ)ϵ{vΣF|mv|v1}N(𝔇Δ)12+2ϵ|\textstyle{\int}_{{\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times}}\varphi_{\xi}^{0}\bigl{(}\iota_{\Delta}(\tau)R_{\Delta}^{-1}\bigr{)}\,{{d}}^{\times}\tau|\ll{\operatorname{vol}}({\mathbb{A}}^{\times}E^{\times}\backslash{\mathbb{A}}_{E}^{\times};{{d}}^{\times}\tau)\ll_{\epsilon}\{\prod_{v\in\Sigma_{F}}|m_{v}|_{v}^{-1}\}{\operatorname{N}}({\mathfrak{D}}_{\Delta})^{\frac{1}{2}+2\epsilon}, which should be a substitute of Lemma 7.8. By Theorem 7.9 and the arguments in § 7.4, we have the absolute convergence and the desired formula. ∎

Theorem 9.1 will be applied to non-vanishing of L𝐿L-values for GL(2)×GL(3)GL2GL3{\operatorname{GL}}(2)\times{\operatorname{GL}}(3) in our forthcoming paper [28].

10. Explicit formulas of local orbital integrals

In this section, we prove Theorems 6.4 and 7.9 by separating cases as in the following table.

v𝑣v Ev/Fvsubscript𝐸𝑣subscript𝐹𝑣E_{v}/F_{v} Proof
ΣsubscriptΣ\Sigma_{\infty} split §10.1.1
ΣsubscriptΣ\Sigma_{\infty} ramified §10.2.1
Σfin(SS(𝔫))subscriptΣfin𝑆𝑆𝔫\Sigma_{{\rm fin}}-(S\cup S({\mathfrak{n}})) split §10.1.2
Σfin(SS(𝔫))subscriptΣfin𝑆𝑆𝔫\Sigma_{{\rm fin}}-(S\cup S({\mathfrak{n}})) non-split §10.2.2
S(𝔫)𝑆𝔫S({\mathfrak{n}}) split §10.1.3
S(𝔫)𝑆𝔫S({\mathfrak{n}}) non-split §10.2.3
S𝑆S split §10.1.4
S𝑆S non-split §10.2.4

To ease notation, in the archimedean cases (§10.1.1 and 10.2.1), we write l𝑙l and |||\,| for lvsubscript𝑙𝑣l_{v} and ||v|\,|_{v}, respectively. In the non-archimedean cases (§10.1.2, §10.1.3, §10.1.4, §10.2.2, §10.2.3 and §10.2.4), we omit the subscript v𝑣v of 𝔬vsubscript𝔬𝑣\mathfrak{o}_{v}, 𝔭vsubscript𝔭𝑣{\mathfrak{p}}_{v}, qvsubscript𝑞𝑣q_{v}, dvsubscript𝑑𝑣d_{v} and ||v|\,|_{v} and write them 𝔬𝔬\mathfrak{o}, 𝔭𝔭{\mathfrak{p}}, q𝑞q, d𝑑d and |||\,|, respectively.

10.1. Local hyperbolic orbital integrals

Let vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}. In this subsection, we compute the hyperbolic local orbital integral

𝔉v(z)(a)=Fv(𝕂vΦv(k1[a(a1)x01]k)𝑑k)φv(0,z)([1x01])𝑑x,vΣF,aFv×{1},formulae-sequencesuperscriptsubscript𝔉𝑣𝑧𝑎subscriptsubscript𝐹𝑣subscriptsubscript𝕂𝑣subscriptΦ𝑣superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘differential-d𝑘superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥formulae-sequence𝑣subscriptΣ𝐹𝑎superscriptsubscript𝐹𝑣1{\mathfrak{F}}_{v}^{(z)}(a)=\int_{F_{v}}\left(\int_{{\mathbb{K}}_{v}}\Phi_{v}\left(k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)\,{{d}}k\right)\,\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x,\quad v\in\Sigma_{F},\quad a\in F_{v}^{\times}-\{1\},

where Φv(gv)subscriptΦ𝑣subscript𝑔𝑣\Phi_{v}(g_{v}) is the v𝑣v-th factor of Φ(𝐬;g)Φ𝐬𝑔\Phi({\mathbf{s}};g).

10.1.1. The proof of Theorem 6.4 (1)

Suppose vΣ𝑣subscriptΣv\in\Sigma_{\infty}, so that Fvsubscript𝐹𝑣F_{v}\cong\mathbb{R}. Fix aFv×{1}𝑎superscriptsubscript𝐹𝑣1a\in F_{v}^{\times}-\{1\}.

Lemma 10.1.

On the region |Re(z)|<1Re𝑧1|\operatorname{Re}(z)|<1, we have

𝔉v(z)(a)=Av(0,z)Jv(z)(a)+Av(0,z)Jv(z)(a),superscriptsubscript𝔉𝑣𝑧𝑎subscript𝐴𝑣0𝑧superscriptsubscript𝐽𝑣𝑧𝑎subscript𝐴𝑣0𝑧superscriptsubscript𝐽𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a)=A_{v}(0,z)J_{v}^{(z)}(a)+A_{v}(0,-z)\,J_{v}^{(-z)}(a),

where Jv(z)(a)superscriptsubscript𝐽𝑣𝑧𝑎J_{v}^{(z)}(a) for Re(z)>1Re𝑧1\operatorname{Re}(z)>-1 is defined as

(10.1) δ(a>0) 2lπ1/2Γ(z2+1)Γ(l12+z+14)Γ(l2+z+14)Γ(z+14)2Γ(z+34)Γ(l+z+14)𝛿𝑎0superscript2𝑙superscript𝜋12Γ𝑧21Γ𝑙12𝑧14Γ𝑙2𝑧14Γsuperscript𝑧142Γ𝑧34Γ𝑙𝑧14\displaystyle\delta(a>0)\,2^{l}\pi^{1/2}\frac{\Gamma\left(\tfrac{z}{2}+1\right)\Gamma\left(\tfrac{l-1}{2}+\tfrac{z+1}{4}\right)\Gamma\left(\tfrac{l}{2}+\tfrac{z+1}{4}\right)}{\Gamma\left(\tfrac{z+1}{4}\right)^{2}\Gamma\left(\tfrac{z+3}{4}\right)\Gamma\left(l+\tfrac{z+1}{4}\right)}
×|a|l/2|a+1|l01(1y)z+141F12(l+12,l2;l+z+14;1|a1a+1|2y)𝑑y.absentsuperscript𝑎𝑙2superscript𝑎1𝑙superscriptsubscript01superscript1𝑦𝑧141subscriptsubscript𝐹12𝑙12𝑙2𝑙𝑧141superscript𝑎1𝑎12𝑦differential-d𝑦\displaystyle\times|a|^{l/2}|a+1|^{-l}\int_{0}^{1}(1-y)^{\frac{z+1}{4}-1}{}_{2}F_{1}\left(\tfrac{l+1}{2},\tfrac{l}{2};l+\tfrac{z+1}{4};1-\left|\tfrac{a-1}{a+1}\right|^{2}\,y\right)\,{{d}}y.
Proof.

Set

Jv(z)(a)=𝕂vΦvl(k1[a(a1)x01]k)hv(0,z)(x)𝑑x𝑑k.superscriptsubscript𝐽𝑣𝑧𝑎subscriptsubscript𝕂𝑣subscriptsuperscriptsubscriptΦ𝑣𝑙superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘superscriptsubscript𝑣0𝑧𝑥differential-d𝑥differential-d𝑘J_{v}^{(z)}(a)=\textstyle{\int}_{{\mathbb{K}}_{v}}\textstyle{\int}_{{\mathbb{R}}}\Phi_{v}^{l}\left(k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)\,h_{v}^{(0,z)}\left(x\right)\,{{d}}x\,{{d}}k.

By the Cartan decomposition, we have

Φvl(k1[a(a1)x01]k)=δ(a>0) 2lal/2{(a+1)+i(a1)x}lsuperscriptsubscriptΦ𝑣𝑙superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘𝛿𝑎0superscript2𝑙superscript𝑎𝑙2superscript𝑎1𝑖𝑎1𝑥𝑙\Phi_{v}^{l}\left(k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)=\delta(a>0)\,2^{l}a^{l/2}\,\{(a+1)+i(a-1)x\}^{-l}

for any k𝕂v𝑘subscript𝕂𝑣k\in{\mathbb{K}}_{v}, a×𝑎superscripta\in{\mathbb{R}}^{\times} and x𝑥x\in{\mathbb{R}}. Since |h(0,z)(x)||logx|much-less-thansuperscript0𝑧𝑥𝑥|h^{(0,z)}(x)|\ll|\log x| as x0𝑥0x\rightarrow 0 and |h(0,z)(x)|(1+x2)Re(z)+14much-less-thansuperscript0𝑧𝑥superscript1superscript𝑥2Re𝑧14|h^{(0,z)}(x)|\ll(1+x^{2})^{-\frac{\operatorname{Re}(z)+1}{4}} as |x|𝑥|x|\rightarrow\infty, the integral Jv(z)(a)superscriptsubscript𝐽𝑣𝑧𝑎J_{v}^{(z)}(a) converges absolutely for Re(z)>12lRe𝑧12𝑙\operatorname{Re}(z)>1-2l and defines a holomorphic function. From now on, we suppose a>0𝑎0a>0 and a1𝑎1a\neq 1. Then,

Jv(z)(a)superscriptsubscript𝐽𝑣𝑧𝑎\displaystyle J_{v}^{(z)}(a) =2lal/2{(a+1)i(a1)x}l(1+x2)z+14F12(z+14,z+14;z2+1;11+x2)𝑑xabsentsuperscript2𝑙superscript𝑎𝑙2subscriptsuperscript𝑎1𝑖𝑎1𝑥𝑙superscript1superscript𝑥2𝑧14subscriptsubscript𝐹12𝑧14𝑧14𝑧2111superscript𝑥2differential-d𝑥\displaystyle=2^{l}a^{l/2}\textstyle{\int}_{{\mathbb{R}}}\{(a+1)-i(a-1)x\}^{-l}(1+x^{2})^{-\frac{z+1}{4}}{}_{2}F_{1}\left(\tfrac{z+1}{4},\tfrac{z+1}{4};\tfrac{z}{2}+1;\tfrac{1}{1+x^{2}}\right)\,{{d}}x
=2lal/2Γ(z2+1)Γ(z+14)2m=0Γ(z+14+m)2m!Γ(z2+m+1){(a+1)i(a1)x}l(1+x2)z+14m𝑑x,absentsuperscript2𝑙superscript𝑎𝑙2Γ𝑧21Γsuperscript𝑧142superscriptsubscript𝑚0Γsuperscript𝑧14𝑚2𝑚Γ𝑧2𝑚1subscriptsuperscript𝑎1𝑖𝑎1𝑥𝑙superscript1superscript𝑥2𝑧14𝑚differential-d𝑥\displaystyle=2^{l}a^{l/2}\frac{\Gamma\left(\frac{z}{2}+1\right)}{\Gamma\left(\tfrac{z+1}{4}\right)^{2}}\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{z+1}{4}+m\right)^{2}}{m!\,\Gamma\left(\tfrac{z}{2}+m+1\right)}\textstyle{\int}_{{\mathbb{R}}}\{(a+1)-i(a-1)x\}^{-l}(1+x^{2})^{-\frac{z+1}{4}-m}\,{{d}}x,

which is valid for z{0,2,4,}𝑧024z\notin\{0,-2,-4,\ldots\}. For any Re(α)>1/2Re𝛼12\operatorname{Re}(\alpha)>1/2,

{(a+1)i(a1)x}l(1+x2)α𝑑xsubscriptsuperscript𝑎1𝑖𝑎1𝑥𝑙superscript1superscript𝑥2𝛼differential-d𝑥\displaystyle\textstyle{\int}_{{\mathbb{R}}}\{(a+1)-i(a-1)x\}^{-l}(1+x^{2})^{-\alpha}\,{{d}}x
=\displaystyle= Γ(l)10e{(a+1)i(a1)x}ttl1𝑑t(1+x2)α𝑑xsubscriptΓsuperscript𝑙1superscriptsubscript0superscript𝑒𝑎1𝑖𝑎1𝑥𝑡superscript𝑡𝑙1differential-d𝑡superscript1superscript𝑥2𝛼differential-d𝑥\displaystyle\textstyle{\int}_{{\mathbb{R}}}{\Gamma(l)^{-1}}\int_{0}^{\infty}e^{-\{(a+1)-i(a-1)x\}t}t^{l-1}{{d}}t\,(1+x^{2})^{-\alpha}\,{{d}}x
=\displaystyle= Γ(l)10e(a+1)ttl1(e(a1)itx(1+x2)α𝑑x)𝑑tΓsuperscript𝑙1superscriptsubscript0superscript𝑒𝑎1𝑡superscript𝑡𝑙1subscriptsuperscript𝑒𝑎1𝑖𝑡𝑥superscript1superscript𝑥2𝛼differential-d𝑥differential-d𝑡\displaystyle{\Gamma(l)^{-1}}\textstyle{\int}_{0}^{\infty}e^{-(a+1)t}t^{l-1}\left(\textstyle{\int}_{{\mathbb{R}}}e^{(a-1)itx}(1+x^{2})^{-\alpha}{{d}}x\right)\,{{d}}t
=\displaystyle= 2π1/2(|a1|2)α1/2Γ(l)1Γ(α)10tl+α3/2e(a+1)tKα1/2(|a1|t)𝑑t2superscript𝜋12superscript𝑎12𝛼12Γsuperscript𝑙1Γsuperscript𝛼1superscriptsubscript0superscript𝑡𝑙𝛼32superscript𝑒𝑎1𝑡subscript𝐾𝛼12𝑎1𝑡differential-d𝑡\displaystyle 2\pi^{1/2}\left(\tfrac{|a-1|}{2}\right)^{\alpha-1/2}\Gamma(l)^{-1}\Gamma(\alpha)^{-1}\textstyle{\int}_{0}^{\infty}t^{l+\alpha-3/2}e^{-(a+1)t}K_{\alpha-1/2}(|a-1|t)\,{{d}}t

by the formula in the last line of [16, p.85]. From now on suppose Re(z)>1Re𝑧1\operatorname{Re}(z)>1. By this, we see that Jv(z)(a)superscriptsubscript𝐽𝑣𝑧𝑎J_{v}^{(z)}(a) for Re(z)>1Re𝑧1\operatorname{Re}(z)>1 is the product of

(10.2) 2l+1al/2π1/2Γ(z2+1)Γ(l)1Γ(z+14)2superscript2𝑙1superscript𝑎𝑙2superscript𝜋12Γ𝑧21Γsuperscript𝑙1Γsuperscript𝑧142\displaystyle{2^{l+1}a^{l/2}\pi^{1/2}\,\Gamma\left(\tfrac{z}{2}+1\right)}{\Gamma(l)^{-1}\,\Gamma\left(\tfrac{z+1}{4}\right)^{-2}}

and

(10.3) m=0Γ(z+14+m)m!Γ(z2+1+m)(|a1|2)z+14+m1/20tl+z+14+m32e(a+1)tKz+14+m12(|a1|t)𝑑t.superscriptsubscript𝑚0Γ𝑧14𝑚𝑚Γ𝑧21𝑚superscript𝑎12𝑧14𝑚12superscriptsubscript0superscript𝑡𝑙𝑧14𝑚32superscript𝑒𝑎1𝑡subscript𝐾𝑧14𝑚12𝑎1𝑡differential-d𝑡\displaystyle\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{z+1}{4}+m\right)}{m!\,\Gamma\left(\tfrac{z}{2}+1+m\right)}\left(\tfrac{|a-1|}{2}\right)^{\tfrac{z+1}{4}+m-1/2}\textstyle{\int}_{0}^{\infty}t^{l+\frac{z+1}{4}+m-\frac{3}{2}}e^{-(a+1)t}K_{\frac{z+1}{4}+m-\frac{1}{2}}(|a-1|t)\,{{d}}t.

By Kα(z)=120exp(z2(y+y1))yα1𝑑ysubscript𝐾𝛼𝑧12superscriptsubscript0𝑧2𝑦superscript𝑦1superscript𝑦𝛼1differential-d𝑦K_{\alpha}(z)=\frac{1}{2}\int_{0}^{\infty}\exp\left(-\tfrac{z}{2}(y+y^{-1})\right)\,y^{-\alpha-1}{{d}}y, the formula (10.3) becomes

(|a1|2)z+1412m=0Γ(z+14+m)m!Γ(z2+1+m)(|a1|2)m0tl+z+14+m32e(a+1)t𝑑tsuperscript𝑎12𝑧1412superscriptsubscript𝑚0Γ𝑧14𝑚𝑚Γ𝑧21𝑚superscript𝑎12𝑚superscriptsubscript0superscript𝑡𝑙𝑧14𝑚32superscript𝑒𝑎1𝑡differential-d𝑡\displaystyle\left(\tfrac{|a-1|}{2}\right)^{\frac{z+1}{4}-\frac{1}{2}}\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{z+1}{4}+m\right)}{m!\,\Gamma\left(\tfrac{z}{2}+1+m\right)}\left(\tfrac{|a-1|}{2}\right)^{m}\textstyle{\int}_{0}^{\infty}t^{l+\frac{z+1}{4}+m-\frac{3}{2}}e^{-(a+1)t}\,{{d}}t\,
×120exp(|a1|t2(y+y1))yz+14m12𝑑yabsent12superscriptsubscript0𝑎1𝑡2𝑦superscript𝑦1superscript𝑦𝑧14𝑚12differential-d𝑦\displaystyle\times\tfrac{1}{2}\textstyle{\int}_{0}^{\infty}\exp\left(-\tfrac{|a-1|t}{2}(y+y^{-1})\right)y^{-\frac{z+1}{4}-m-\frac{1}{2}}{{d}}y
=\displaystyle= 12(|a1|2)z+1412Γ(z+14)Γ(z2+1)100𝑑t𝑑ytl+z+1432e(a+1)t12superscript𝑎12𝑧1412Γ𝑧14Γsuperscript𝑧211superscriptsubscript0superscriptsubscript0differential-d𝑡differential-d𝑦superscript𝑡𝑙𝑧1432superscript𝑒𝑎1𝑡\displaystyle\tfrac{1}{2}\left(\tfrac{|a-1|}{2}\right)^{\frac{z+1}{4}-\frac{1}{2}}{\Gamma\left(\tfrac{z+1}{4}\right)}{\Gamma\left(\tfrac{z}{2}+1\right)^{-1}}\textstyle{\int}_{0}^{\infty}\int_{0}^{\infty}{{d}}t\,{{d}}y\,t^{l+\frac{z+1}{4}-\frac{3}{2}}e^{-(a+1)t}
×exp(|a1|t2(y+y1))F11(z+14,z2+1;ty|a1|2)yz+1412absent𝑎1𝑡2𝑦superscript𝑦1subscriptsubscript𝐹11𝑧14𝑧21𝑡𝑦𝑎12superscript𝑦𝑧1412\displaystyle\times\exp\left(-\tfrac{|a-1|t}{2}(y+y^{-1})\right){}_{1}F_{1}\left(\tfrac{z+1}{4},\tfrac{z}{2}+1;\tfrac{t}{y}\tfrac{|a-1|}{2}\right)y^{-\frac{z+1}{4}-\frac{1}{2}}
=\displaystyle= 12(|a1|2)z+1412Γ(z+14)Γ(z2+1)100𝑑t𝑑ytl+z+1432e(a+1)yt12superscript𝑎12𝑧1412Γ𝑧14Γsuperscript𝑧211superscriptsubscript0superscriptsubscript0differential-d𝑡differential-d𝑦superscript𝑡𝑙𝑧1432superscript𝑒𝑎1𝑦𝑡\displaystyle\tfrac{1}{2}\left(\tfrac{|a-1|}{2}\right)^{\frac{z+1}{4}-\frac{1}{2}}{\Gamma\left(\tfrac{z+1}{4}\right)}{\Gamma\left(\tfrac{z}{2}+1\right)^{-1}}\textstyle{\int}_{0}^{\infty}\int_{0}^{\infty}{{d}}t\,{{d}}y\,t^{l+\frac{z+1}{4}-\frac{3}{2}}e^{-(a+1)yt}
×exp(|a1|yt2(y+y1))F11(z+14,z2+1;t|a1|2)yl1.absent𝑎1𝑦𝑡2𝑦superscript𝑦1subscriptsubscript𝐹11𝑧14𝑧21𝑡𝑎12superscript𝑦𝑙1\displaystyle\times\exp\left(-\tfrac{|a-1|yt}{2}(y+y^{-1})\right){}_{1}F_{1}\left(\tfrac{z+1}{4},\tfrac{z}{2}+1;\tfrac{t|a-1|}{2}\right)y^{l-1}.

To have the last equality, we made the variable change tty𝑡𝑡𝑦t\rightarrow ty. The y𝑦y-integral is computed as

0exp(|a1|2ty2(a+1)ty)yl1𝑑ysuperscriptsubscript0𝑎12𝑡superscript𝑦2𝑎1𝑡𝑦superscript𝑦𝑙1differential-d𝑦\displaystyle\textstyle{\int}_{0}^{\infty}\exp\left(\tfrac{-|a-1|}{2}ty^{2}-(a+1)ty\right)y^{l-1}\,{{d}}y =(1|a1|t)l/20exp(y22a+1|a1|1/2t1/2y)yl1𝑑yabsentsuperscript1𝑎1𝑡𝑙2superscriptsubscript0superscript𝑦22𝑎1superscript𝑎112superscript𝑡12𝑦superscript𝑦𝑙1differential-d𝑦\displaystyle=\left(\tfrac{1}{|a-1|t}\right)^{l/2}\textstyle{\int}_{0}^{\infty}\exp\left(-\tfrac{y^{2}}{2}-\tfrac{a+1}{|a-1|^{1/2}}t^{1/2}y\right)y^{l-1}\,{{d}}y
=(1|a1|t)l/2Γ(l)exp((a+1)2t4|a1|)Dl(a+1|a1|1/2t1/2)absentsuperscript1𝑎1𝑡𝑙2Γ𝑙superscript𝑎12𝑡4𝑎1subscript𝐷𝑙𝑎1superscript𝑎112superscript𝑡12\displaystyle=\left(\tfrac{1}{|a-1|t}\right)^{l/2}\Gamma(l)\exp\left(\tfrac{(a+1)^{2}t}{4|a-1|}\right)\,D_{-l}\left(\tfrac{a+1}{|a-1|^{1/2}}t^{1/2}\right)

in terms of the parabolic cylinder function Dl(z)subscript𝐷𝑙𝑧D_{-l}(z) by the first formula on [16, p.328]. Hence,

Jv(z)(a)=superscriptsubscript𝐽𝑣𝑧𝑎absent\displaystyle J_{v}^{(z)}(a)= 2lal/2π1/2Γ(z+14)1|a1|l/2(|a1|2)z+1412superscript2𝑙superscript𝑎𝑙2superscript𝜋12Γsuperscript𝑧141superscript𝑎1𝑙2superscript𝑎12𝑧1412\displaystyle{2^{l}a^{l/2}{\pi^{1/2}}}{\Gamma\left(\tfrac{z+1}{4}\right)^{-1}}|a-1|^{-l/2}\left(\tfrac{|a-1|}{2}\right)^{\frac{z+1}{4}-\frac{1}{2}}
×0exp((a+1)2t4|a1||a1|t2)Dl(a+1|a1|1/2t1/2)F11(z+14,z2+1;t|a1|2)tl2+z+1432dt\displaystyle\times\textstyle{\int}_{0}^{\infty}\exp\left(\tfrac{(a+1)^{2}t}{4|a-1|}-\tfrac{|a-1|t}{2}\right)\,D_{-l}\left(\tfrac{a+1}{|a-1|^{1/2}}t^{1/2}\right)\,{}_{1}F_{1}\left(\tfrac{z+1}{4},\tfrac{z}{2}+1;\tfrac{t|a-1|}{2}\right)\,t^{\frac{l}{2}+\frac{z+1}{4}-\frac{3}{2}}\,{{d}}t
=\displaystyle= 2lal/2π1/2Γ(z+14)1|a1|l/2(|a1|2)l/2superscript2𝑙superscript𝑎𝑙2superscript𝜋12Γsuperscript𝑧141superscript𝑎1𝑙2superscript𝑎12𝑙2\displaystyle{2^{l}a^{l/2}{\pi^{1/2}}}{\Gamma\left(\tfrac{z+1}{4}\right)^{-1}}|a-1|^{-l/2}\left(\tfrac{|a-1|}{2}\right)^{-l/2}
×0exp({(a+1)22(a1)21}t)Dl(a+1|a1|2t)F11(z+14,z2+1;t)tl2+z+1432dt\displaystyle\times\textstyle{\int}_{0}^{\infty}\exp\left(\left\{\tfrac{(a+1)^{2}}{2(a-1)^{2}}-1\right\}t\right)\,D_{-l}\left(\tfrac{a+1}{|a-1|}\sqrt{2t}\right)\,{}_{1}F_{1}\left(\tfrac{z+1}{4},\tfrac{z}{2}+1;t\right)\,t^{\frac{l}{2}+\frac{z+1}{4}-\frac{3}{2}}\,{{d}}t
=\displaystyle= 2lal/2π1/2Γ(z+14)1|a1|l/2(|a1|2)l/2superscript2𝑙superscript𝑎𝑙2superscript𝜋12Γsuperscript𝑧141superscript𝑎1𝑙2superscript𝑎12𝑙2\displaystyle{2^{l}a^{l/2}{\pi^{1/2}}}{\Gamma\left(\tfrac{z+1}{4}\right)^{-1}}|a-1|^{-l/2}\left(\tfrac{|a-1|}{2}\right)^{-l/2}
×2l/20etU(l2,12,(a+1a1)2t)F11(z+14,z2+1;t)tl2+z+1432𝑑tabsentsuperscript2𝑙2superscriptsubscript0superscript𝑒𝑡𝑈𝑙212superscript𝑎1𝑎12𝑡subscriptsubscript𝐹11𝑧14𝑧21𝑡superscript𝑡𝑙2𝑧1432differential-d𝑡\displaystyle\times 2^{-l/2}\textstyle{\int}_{0}^{\infty}e^{-t}\,U\left(\tfrac{l}{2},\tfrac{1}{2},\left(\tfrac{a+1}{a-1}\right)^{2}t\right)\,{}_{1}F_{1}\left(\tfrac{z+1}{4},\tfrac{z}{2}+1;t\right)\,t^{\frac{l}{2}+\frac{z+1}{4}-\frac{3}{2}}\,{{d}}t

by the formula on [16, p.287]. Applying the integral expression

etF11(z+14,z2+1;t)=Γ(z2+1)Γ(z+14)1Γ(z+34)101etyyz+341(1y)z+141𝑑ysuperscript𝑒𝑡subscriptsubscript𝐹11𝑧14𝑧21𝑡Γ𝑧21Γsuperscript𝑧141Γsuperscript𝑧341superscriptsubscript01superscript𝑒𝑡𝑦superscript𝑦𝑧341superscript1𝑦𝑧141differential-d𝑦e^{-t}{}_{1}F_{1}\left(\tfrac{z+1}{4},\tfrac{z}{2}+1;t\right)={\Gamma\left(\tfrac{z}{2}+1\right)}{\Gamma\left(\tfrac{z+1}{4}\right)^{-1}\Gamma\left(\tfrac{z+3}{4}\right)^{-1}}\textstyle{\int}_{0}^{1}e^{-ty}y^{\frac{z+3}{4}-1}(1-y)^{\frac{z+1}{4}-1}\,{{d}}y

obtained from the formula on the last line of [16, p.274] by an obvious variable change, and then using the formula

0tl2+z+1432etyU(l2,12,βt)𝑑tsuperscriptsubscript0superscript𝑡𝑙2𝑧1432superscript𝑒𝑡𝑦𝑈𝑙212𝛽𝑡differential-d𝑡\displaystyle\textstyle{\int}_{0}^{\infty}t^{\frac{l}{2}+\frac{z+1}{4}-\frac{3}{2}}e^{-ty}U\left(\tfrac{l}{2},\tfrac{1}{2},\beta t\right){{d}}t =βl2z+14+12Γ(l2+z+1412)Γ(l2+z+14)Γ(l+z+14)absentsuperscript𝛽𝑙2𝑧1412Γ𝑙2𝑧1412Γ𝑙2𝑧14Γ𝑙𝑧14\displaystyle=\beta^{-\frac{l}{2}-\frac{z+1}{4}+\frac{1}{2}}\,\frac{\Gamma\left(\frac{l}{2}+\frac{z+1}{4}-\frac{1}{2}\right)\Gamma\left(\frac{l}{2}+\frac{z+1}{4}\right)}{\Gamma\left(l+\frac{z+1}{4}\right)}
×F12(l2+z+1412,l2+z+14;l+z+14;1yβ),absentsubscriptsubscript𝐹12𝑙2𝑧1412𝑙2𝑧14𝑙𝑧141𝑦𝛽\displaystyle\times{}_{2}F_{1}\left(\tfrac{l}{2}+\tfrac{z+1}{4}-\tfrac{1}{2},\tfrac{l}{2}+\tfrac{z+1}{4};l+\tfrac{z+1}{4};1-\tfrac{y}{\beta}\right),

which is deduced from the first formula of [16, §7.5.2] by applying U(a,c;z)=ez/2zc/2Wc/2a,c/21/2(z)𝑈𝑎𝑐𝑧superscript𝑒𝑧2superscript𝑧𝑐2subscript𝑊𝑐2𝑎𝑐212𝑧U(a,c;z)=e^{z/2}z^{-c/2}W_{c/2-a,c/2-1/2}(z) ([16, p.304]), it turns out that Jv(z)(a)superscriptsubscript𝐽𝑣𝑧𝑎J_{v}^{(z)}(a) is the product of

2lπ1/2Γ(z2+1)Γ(l12+z+14)Γ(l2+z+14)Γ(z+14)2Γ(z+34)1Γ(l+z+14)1superscript2𝑙superscript𝜋12Γ𝑧21Γ𝑙12𝑧14Γ𝑙2𝑧14Γsuperscript𝑧142Γsuperscript𝑧341Γsuperscript𝑙𝑧1412^{l}\pi^{1/2}{\Gamma\left(\tfrac{z}{2}+1\right)\Gamma\left(\tfrac{l-1}{2}+\tfrac{z+1}{4}\right)\Gamma\left(\tfrac{l}{2}+\tfrac{z+1}{4}\right)}{\Gamma\left(\tfrac{z+1}{4}\right)^{-2}\Gamma\left(\tfrac{z+3}{4}\right)^{-1}\Gamma\left(l+\tfrac{z+1}{4}\right)^{-1}}

and

|a|l/2|a1|lλ(a)l2+z+141201yz+341(1y)z+141F12(l12+z+14,l2+z+14;l+z+14;1λ(a)y)𝑑y,superscript𝑎𝑙2superscript𝑎1𝑙𝜆superscript𝑎𝑙2𝑧1412superscriptsubscript01superscript𝑦𝑧341superscript1𝑦𝑧141subscriptsubscript𝐹12𝑙12𝑧14𝑙2𝑧14𝑙𝑧141𝜆𝑎𝑦differential-d𝑦|a|^{l/2}|a-1|^{-l}\lambda(a)^{\frac{l}{2}+\frac{z+1}{4}-\frac{1}{2}}\textstyle{\int}_{0}^{1}y^{\frac{z+3}{4}-1}(1-y)^{\frac{z+1}{4}-1}{}_{2}F_{1}\left(\tfrac{l-1}{2}+\tfrac{z+1}{4},\tfrac{l}{2}+\tfrac{z+1}{4};l+\tfrac{z+1}{4};1-\lambda(a)y\right)\,{{d}}y,

where λ(a)=|(a1)/(a+1)|2𝜆𝑎superscript𝑎1𝑎12\lambda(a)=|(a-1)/(a+1)|^{2}. We note that the integral is convergent for Re(z)>1Re𝑧1\operatorname{Re}(z)>-1. We apply the formula F12(a,b;c;z)=(1z)cabF12(ca,cb;c;z)subscriptsubscript𝐹12𝑎𝑏𝑐𝑧superscript1𝑧𝑐𝑎𝑏subscriptsubscript𝐹12𝑐𝑎𝑐𝑏𝑐𝑧{}_{2}F_{1}(a,b;c;z)=(1-z)^{c-a-b}{}_{2}F_{1}(c-a,c-b;c;z) to obtain the desired formula (10.1) for Re(z)>1Re𝑧1\operatorname{Re}(z)>1. Then the equality (10.1) is extended to Re(z)>1Re𝑧1\operatorname{Re}(z)>-1 due to the holomorphy. ∎

Let us return to prove Theorem 6.4 (1). Suppose |Re(z)|<1Re𝑧1|\operatorname{Re}(z)|<1 and set λ=λ(a)𝜆𝜆𝑎\lambda=\lambda(a); we note 0<λ<10𝜆10<\lambda<1 if a>0𝑎0a>0. By the Gauss connection formula for (1z)/41𝑧4(1-z)/4\notin\mathbb{Z} (on the last line of [16, p.47]), the integral

01(1y)z+141F12(l+12,l2;l+z+14;1λy)𝑑ysuperscriptsubscript01superscript1𝑦𝑧141subscriptsubscript𝐹12𝑙12𝑙2𝑙𝑧141𝜆𝑦differential-d𝑦\displaystyle\textstyle{\int}_{0}^{1}(1-y)^{\frac{z+1}{4}-1}{}_{2}F_{1}\left(\tfrac{l+1}{2},\tfrac{l}{2};l+\tfrac{z+1}{4};1-\lambda{y}\right)\,{{d}}y

becomes the sum of

(10.4) Γ(l+z+14)Γ(z14)Γ(z+14+l12)Γ(z+14+l2)01(1y)z34F12(l+12,l2;5z4;λy)𝑑yΓ𝑙𝑧14Γ𝑧14Γ𝑧14𝑙12Γ𝑧14𝑙2superscriptsubscript01superscript1𝑦𝑧34subscriptsubscript𝐹12𝑙12𝑙25𝑧4𝜆𝑦differential-d𝑦\displaystyle\frac{\Gamma\left(l+\tfrac{z+1}{4}\right)\Gamma\left(\tfrac{z-1}{4}\right)}{\Gamma\left(\tfrac{z+1}{4}+\tfrac{l-1}{2}\right)\Gamma\left(\tfrac{z+1}{4}+\tfrac{l}{2}\right)}\textstyle{\int}_{0}^{1}(1-y)^{\frac{z-3}{4}}{}_{2}F_{1}\left(\tfrac{l+1}{2},\tfrac{l}{2};\tfrac{5-z}{4};\lambda y\right)dy

and

(10.5) λz14Γ(l+z+14)Γ(1z4)Γ(l+12)Γ(l2)01yz14(1y)z34F12(z+14+l2,z+14+l12;z+34;λy)𝑑y.superscript𝜆𝑧14Γ𝑙𝑧14Γ1𝑧4Γ𝑙12Γ𝑙2superscriptsubscript01superscript𝑦𝑧14superscript1𝑦𝑧34subscriptsubscript𝐹12𝑧14𝑙2𝑧14𝑙12𝑧34𝜆𝑦differential-d𝑦\displaystyle\lambda^{\frac{z-1}{4}}\,\frac{\Gamma\left(l+\tfrac{z+1}{4}\right)\Gamma\left(\tfrac{1-z}{4}\right)}{\Gamma\left(\tfrac{l+1}{2}\right)\Gamma\left(\tfrac{l}{2}\right)}\textstyle{\int}_{0}^{1}y^{\frac{z-1}{4}}(1-y)^{\frac{z-3}{4}}{}_{2}F_{1}\left(\tfrac{z+1}{4}+\tfrac{l}{2},\tfrac{z+1}{4}+\tfrac{l-1}{2};\tfrac{z+3}{4};\lambda y\right){{d}}y.

We see that these are computed as

(10.4)=italic-(10.4italic-)absent\displaystyle\eqref{FHypL10-1}= Γ(l+z+14)Γ(z14)Γ(z+14+l12)Γ(z+14+l2)Γ(5z4)Γ(z+14)Γ(l2)Γ(l+12)m=0Γ(l2+m)Γ(l+12+m)Γ(z+14+m+1)Γ(5z4+m)λmΓ𝑙𝑧14Γ𝑧14Γ𝑧14𝑙12Γ𝑧14𝑙2Γ5𝑧4Γ𝑧14Γ𝑙2Γ𝑙12superscriptsubscript𝑚0Γ𝑙2𝑚Γ𝑙12𝑚Γ𝑧14𝑚1Γ5𝑧4𝑚superscript𝜆𝑚\displaystyle\frac{\Gamma\left(l+\tfrac{z+1}{4}\right)\Gamma\left(\tfrac{z-1}{4}\right)}{\Gamma\left(\tfrac{z+1}{4}+\tfrac{l-1}{2}\right)\Gamma\left(\tfrac{z+1}{4}+\tfrac{l}{2}\right)}\frac{\Gamma\left(\tfrac{5-z}{4}\right)\Gamma\left(\tfrac{z+1}{4}\right)}{\Gamma\left(\tfrac{l}{2}\right)\Gamma\left(\tfrac{l+1}{2}\right)}\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{l}{2}+m\right)\Gamma\left(\tfrac{l+1}{2}+m\right)}{\Gamma\left(\tfrac{z+1}{4}+m+1\right)\Gamma\left(\tfrac{5-z}{4}+m\right)}\,\lambda^{m}

and

(10.5)=italic-(10.5italic-)absent\displaystyle\eqref{FHypL10-2}= λz14Γ(l+z+14)Γ(1z4)Γ(l+12)Γ(l2)Γ(z+34)Γ(z+14)Γ(z+14+l2)Γ(z+14+l12)m=0Γ(z+14+l2+m)Γ(z+14+l12+m)m!Γ(z2+1+m)λm.superscript𝜆𝑧14Γ𝑙𝑧14Γ1𝑧4Γ𝑙12Γ𝑙2Γ𝑧34Γ𝑧14Γ𝑧14𝑙2Γ𝑧14𝑙12superscriptsubscript𝑚0Γ𝑧14𝑙2𝑚Γ𝑧14𝑙12𝑚𝑚Γ𝑧21𝑚superscript𝜆𝑚\displaystyle\lambda^{\frac{z-1}{4}}\,\frac{\Gamma\left(l+\tfrac{z+1}{4}\right)\Gamma\left(\tfrac{1-z}{4}\right)}{\Gamma\left(\tfrac{l+1}{2}\right)\Gamma\left(\tfrac{l}{2}\right)}\frac{\Gamma\left(\tfrac{z+3}{4}\right)\Gamma\left(\tfrac{z+1}{4}\right)}{\Gamma\left(\tfrac{z+1}{4}+\tfrac{l}{2}\right)\Gamma\left(\tfrac{z+1}{4}+\tfrac{l-1}{2}\right)}\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{z+1}{4}+\tfrac{l}{2}+m\right)\Gamma\left(\tfrac{z+1}{4}+\tfrac{l-1}{2}+m\right)}{m!\,\Gamma\left(\tfrac{z}{2}+1+m\right)}\lambda^{m}.

Combining these evaluations with the formula of Jv(z)(a)superscriptsubscript𝐽𝑣𝑧𝑎J_{v}^{(z)}(a) in Lemma 10.1, we obtain

Jv(z)(a)=superscriptsubscript𝐽𝑣𝑧𝑎absent\displaystyle J_{v}^{(z)}(a)=\, δ(a>0)2lπ1/2Γ(z2+1)Γ(l2)1Γ(l+12)1Γ(z+14)1Γ(z+34)1|a|l/2|a1|l𝛿𝑎0superscript2𝑙superscript𝜋12Γ𝑧21Γsuperscript𝑙21Γsuperscript𝑙121Γsuperscript𝑧141Γsuperscript𝑧341superscript𝑎𝑙2superscript𝑎1𝑙\displaystyle\delta(a>0){2^{l}\pi^{1/2}\Gamma\left(\tfrac{z}{2}+1\right)}{\Gamma\left(\tfrac{l}{2}\right)^{-1}\Gamma\left(\tfrac{l+1}{2}\right)^{-1}\Gamma\left(\tfrac{z+1}{4}\right)^{-1}\Gamma\left(\tfrac{z+3}{4}\right)^{-1}}|a|^{l/2}|a-1|^{-l}
×{Γ(z14)Γ(5z4)m=0Γ(l2+m)Γ(l+12+m)Γ(5+z4+m)Γ(5z4+m)λm\displaystyle\times\biggl{\{}\Gamma\left(\tfrac{z-1}{4}\right)\Gamma\left(\tfrac{5-z}{4}\right)\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{l}{2}+m\right)\Gamma\left(\tfrac{l+1}{2}+m\right)}{\Gamma\left(\tfrac{5+z}{4}+m\right)\Gamma\left(\tfrac{5-z}{4}+m\right)}\,\lambda^{m}
+λz14Γ(z+34)Γ(1z4)m=0Γ(z+14+l2+m)Γ(z+14+l12+m)m!Γ(z2+1+m)λm}\displaystyle\qquad+\lambda^{\frac{z-1}{4}}\Gamma\left(\tfrac{z+3}{4}\right)\Gamma\left(\tfrac{1-z}{4}\right)\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{z+1}{4}+\tfrac{l}{2}+m\right)\Gamma\left(\tfrac{z+1}{4}+\tfrac{l-1}{2}+m\right)}{m!\,\Gamma\left(\tfrac{z}{2}+1+m\right)}\lambda^{m}\biggr{\}}
=\displaystyle=\, δ(a>0)22l1Γ(z2+1)Γ(1z4)Γ(l)1Γ(z+14)1|a|l/2|a1|l𝛿𝑎0superscript22𝑙1Γ𝑧21Γ1𝑧4Γsuperscript𝑙1Γsuperscript𝑧141superscript𝑎𝑙2superscript𝑎1𝑙\displaystyle\delta(a>0){2^{2l-1}\Gamma\left(\tfrac{z}{2}+1\right)\Gamma\left(\tfrac{1-z}{4}\right)}{\Gamma\left({l}\right)^{-1}\Gamma\left(\tfrac{z+1}{4}\right)^{-1}}|a|^{l/2}|a-1|^{-l}
×{m=0Γ(l2+m)Γ(l+12+m)Γ(5+z4+m)Γ(5z4+m)λm+λz14m=0Γ(z+14+l2+m)Γ(z14+l2+m)m!Γ(z2+1+m)λm}absentsuperscriptsubscript𝑚0Γ𝑙2𝑚Γ𝑙12𝑚Γ5𝑧4𝑚Γ5𝑧4𝑚superscript𝜆𝑚superscript𝜆𝑧14superscriptsubscript𝑚0Γ𝑧14𝑙2𝑚Γ𝑧14𝑙2𝑚𝑚Γ𝑧21𝑚superscript𝜆𝑚\displaystyle\times\biggl{\{}-\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{l}{2}+m\right)\Gamma\left(\tfrac{l+1}{2}+m\right)}{\Gamma\left(\tfrac{5+z}{4}+m\right)\Gamma\left(\tfrac{5-z}{4}+m\right)}\,\lambda^{m}+\lambda^{\frac{z-1}{4}}\sum_{m=0}^{\infty}\frac{\Gamma\left(\tfrac{z+1}{4}+\tfrac{l}{2}+m\right)\Gamma\left(\tfrac{z-1}{4}+\tfrac{l}{2}+m\right)}{m!\,\Gamma\left(\tfrac{z}{2}+1+m\right)}\lambda^{m}\biggr{\}}

using the formula Γ(z+34)Γ(1z4)=πsin(z+34π)=πsin(z14π)=Γ(z14)Γ(5z4)Γ𝑧34Γ1𝑧4𝜋𝑧34𝜋𝜋𝑧14𝜋Γ𝑧14Γ5𝑧4\Gamma\left(\tfrac{z+3}{4}\right)\Gamma\left(\tfrac{1-z}{4}\right)=\frac{\pi}{\sin(\frac{z+3}{4}\pi)}=\frac{-\pi}{\sin(\frac{z-1}{4}\pi)}=-\Gamma\left(\tfrac{z-1}{4}\right)\Gamma\left(\tfrac{5-z}{4}\right), and the duplication formula Γ(l)=2l1π1/2Γ(l/2)Γ((l+1)/2)Γ𝑙superscript2𝑙1superscript𝜋12Γ𝑙2Γ𝑙12\Gamma(l)=2^{l-1}\pi^{-1/2}\Gamma(l/2)\Gamma((l+1)/2). Pluging this and Av(0,z)=πΓ(z/2)Γ((1z)/4)2subscript𝐴𝑣0𝑧𝜋Γ𝑧2Γsuperscript1𝑧42A_{v}(0,z)=\frac{\sqrt{\pi}\Gamma(-z/2)}{\Gamma((1-z)/4)^{2}} to 𝔉v(z)(a)=Av(0,z)Jv(z)(a)+Av(0,z)Jv(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎subscript𝐴𝑣0𝑧superscriptsubscript𝐽𝑣𝑧𝑎subscript𝐴𝑣0𝑧superscriptsubscript𝐽𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a)=A_{v}(0,z)J_{v}^{(z)}(a)+A_{v}(0,-z)J_{v}^{(-z)}(a), after a simple computation, we have the formula

𝔉v(z)(a)=superscriptsubscript𝔉𝑣𝑧𝑎absent\displaystyle{\mathfrak{F}}_{v}^{(z)}(a)= 22l1πΓ(l)1 2Γ(1+z2)Γ(1+z2)z1Γ(1+z4)1Γ(1z4)1superscript22𝑙1𝜋Γsuperscript𝑙12Γ1𝑧2Γ1𝑧2superscript𝑧1Γsuperscript1𝑧41Γsuperscript1𝑧41\displaystyle 2^{2l-1}\sqrt{\pi}\,\Gamma(l)^{-1}\,{2\Gamma\left(1+\tfrac{z}{2}\right)\Gamma\left(1+\tfrac{-z}{2}\right)}\,{z^{-1}\,\Gamma\left(\tfrac{1+z}{4}\right)^{-1}\Gamma\left(\tfrac{1-z}{4}\right)^{-1}}
×δ(a>0)|a|l/2|a+1|l{λ(a)z14𝒢l(z)(λ(a))+λ(a)z14𝒢l(z)(λ(a))},absent𝛿𝑎0superscript𝑎𝑙2superscript𝑎1𝑙𝜆superscript𝑎𝑧14superscriptsubscript𝒢𝑙𝑧𝜆𝑎𝜆superscript𝑎𝑧14superscriptsubscript𝒢𝑙𝑧𝜆𝑎\displaystyle\times\delta(a>0)\,|a|^{l/2}|a+1|^{-l}\{-\lambda(a)^{\frac{z-1}{4}}{\mathcal{G}}_{l}^{(z)}(\lambda(a))+\lambda(a)^{\frac{-z-1}{4}}{\mathcal{G}}_{l}^{(-z)}\left(\lambda(a)\right)\},

where for z𝑧z\in{\mathbb{C}}, l2𝑙2l\in 2{\mathbb{N}} and |λ|<1𝜆1|\lambda|<1, we set

(10.6) 𝒢l(z)(λ)=Γ(z+14+l2)Γ(z+14+l12)Γ(z2+1)1F12(z+14+l2,z+14+l12;z2+1;λ).superscriptsubscript𝒢𝑙𝑧𝜆Γ𝑧14𝑙2Γ𝑧14𝑙12Γsuperscript𝑧211subscriptsubscript𝐹12𝑧14𝑙2𝑧14𝑙12𝑧21𝜆\displaystyle{\mathcal{G}}_{l}^{(z)}(\lambda)={\Gamma\left(\tfrac{z+1}{4}+\tfrac{l}{2}\right)\Gamma\left(\tfrac{z+1}{4}+\tfrac{l-1}{2}\right)}{\Gamma\left(\tfrac{z}{2}+1\right)^{-1}}\,{}_{2}F_{1}\left(\tfrac{z+1}{4}+\tfrac{l}{2},\tfrac{z+1}{4}+\tfrac{l-1}{2};\tfrac{z}{2}+1;\lambda\right).

From [16, line 14, p.153] and [16, line 18, p.164],

λz14𝒢l(z)(λ)=22l(1λ)1l2λ1/2𝔔z12l1(λ1/2)superscript𝜆𝑧14superscriptsubscript𝒢𝑙𝑧𝜆superscript22𝑙superscript1𝜆1𝑙2superscript𝜆12superscriptsubscript𝔔𝑧12𝑙1superscript𝜆12\displaystyle\lambda^{\frac{z-1}{4}}\,{\mathcal{G}}_{l}^{(z)}(\lambda)=-2^{2-l}(1-\lambda)^{\frac{1-l}{2}}\lambda^{-1/2}\,{\mathfrak{Q}}_{\frac{z-1}{2}}^{l-1}\left(\lambda^{-1/2}\right)

and

𝔔z12l1(λ1/2)𝔔z12l1(λ1/2)=superscriptsubscript𝔔𝑧12𝑙1superscript𝜆12superscriptsubscript𝔔𝑧12𝑙1superscript𝜆12absent\displaystyle{\mathfrak{Q}}_{\frac{z-1}{2}}^{l-1}(\lambda^{-1/2})-{\mathfrak{Q}}_{\frac{-z-1}{2}}^{l-1}(\lambda^{-1/2})= sin(πz2)Γ(l+z12)Γ(l+z12)𝔓z121l(λ1/2),𝜋𝑧2Γ𝑙𝑧12Γ𝑙𝑧12superscriptsubscript𝔓𝑧121𝑙superscript𝜆12\displaystyle\sin\left(\tfrac{\pi z}{2}\right)\,\Gamma\left(l+\tfrac{z-1}{2}\right)\Gamma\left(l+\tfrac{-z-1}{2}\right)\,{\mathfrak{P}}_{\frac{z-1}{2}}^{1-l}(\lambda^{-1/2}),

where 𝔓νμ(x)superscriptsubscript𝔓𝜈𝜇𝑥{\mathfrak{P}}_{\nu}^{\mu}(x) and 𝔔νμ(x)superscriptsubscript𝔔𝜈𝜇𝑥{\mathfrak{Q}}_{\nu}^{\mu}(x) are the associated Legendre functions of the first kind and of the second kind, respectively. Thus we obtain the desired formula for |Re(z)|<1Re𝑧1|\operatorname{Re}(z)|<1. By analytic continuation, it remains valid on |Re(z)|<2l1Re𝑧2𝑙1|\operatorname{Re}(z)|<2l-1. This completes the proof of Theorem 6.4 (1).

10.1.2. The proof of Theorem 6.4 (2)

Let vΣfin(SS(𝔫))𝑣subscriptΣfin𝑆𝑆𝔫v\in\Sigma_{\rm fin}-(S\cup S({\mathfrak{n}})). Recall Φv=chZv𝕂vsubscriptΦ𝑣subscriptchsubscript𝑍𝑣subscript𝕂𝑣\Phi_{v}={\rm ch}_{Z_{v}{\mathbb{K}}_{v}}. For aFv×{1}𝑎superscriptsubscript𝐹𝑣1a\in F_{v}^{\times}-\{1\} and xFv𝑥subscript𝐹𝑣x\in F_{v}, we have Φv(k1[a(a1)x01]k)0subscriptΦ𝑣superscript𝑘1delimited-[]𝑎𝑎1𝑥01𝑘0\Phi_{v}\left(k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right)\not=0 for some k𝕂v𝑘subscript𝕂𝑣k\in{\mathbb{K}}_{v} if and only if a𝔬×𝑎superscript𝔬a\in\mathfrak{o}^{\times} and x(a1)1𝔬𝑥superscript𝑎11𝔬x\in(a-1)^{-1}\mathfrak{o}. Thus, 𝔉v(z)(a)=0superscriptsubscript𝔉𝑣𝑧𝑎0{\mathfrak{F}}_{v}^{(z)}(a)=0 unless a𝔬×𝑎superscript𝔬a\in\mathfrak{o}^{\times}, in which case 𝔉v(z)(a)=Av(0,z)I(z)+Av(0,z)I(z)superscriptsubscript𝔉𝑣𝑧𝑎subscript𝐴𝑣0𝑧𝐼𝑧subscript𝐴𝑣0𝑧𝐼𝑧{\mathfrak{F}}_{v}^{(z)}(a)=A_{v}(0,z)\,I(z)+A_{v}(0,-z)I(-z), where I(z)=x(a1)1𝔬hv(0,z)(x)𝑑x𝐼𝑧subscript𝑥superscript𝑎11𝔬superscriptsubscript𝑣0𝑧𝑥differential-d𝑥I(z)=\int_{x\in(a-1)^{-1}\mathfrak{o}}h_{v}^{(0,z)}(x)\,{{d}}x. For a𝔬×𝑎superscript𝔬a\in\mathfrak{o}^{\times}, we easily have that I(z)𝐼𝑧I(z) is absolutely convergent for all z𝑧z and

I(z)=qd/2{1ζFv(1)1ζFv(z+12)(1|a1|z12)}𝐼𝑧superscript𝑞𝑑21subscript𝜁subscript𝐹𝑣superscript11subscript𝜁subscript𝐹𝑣𝑧121superscript𝑎1𝑧12I(z)=q^{-d/2}\left\{1-\zeta_{F_{v}}(1)^{-1}\zeta_{F_{v}}\left(\tfrac{-z+1}{2}\right)(1-|a-1|^{\frac{z-1}{2}})\right\}

if qvz1superscriptsubscript𝑞𝑣𝑧1q_{v}^{z}\not=1. Hence

𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎\displaystyle{\mathfrak{F}}_{v}^{(z)}(a) =qd/2δ(a𝔬×){Av(0,z)+Av(0,z)\displaystyle=q^{-d/2}\,\delta(a\in\mathfrak{o}^{\times})\,\biggl{\{}A_{v}(0,z)+A_{v}(0,-z)
ζFv(z)ζFv(z+12)1(1|a1|z12)ζFv(z)ζFv(z+12)1(1|a1|z12)}.\displaystyle-{\zeta_{F_{v}}(-z)}{\zeta_{F_{v}}\left(\tfrac{-z+1}{2}\right)^{-1}}(1-|a-1|^{\frac{z-1}{2}})-{\zeta_{F_{v}}(z)}{\zeta_{F_{v}}\left(\tfrac{z+1}{2}\right)^{-1}}(1-|a-1|^{\frac{-z-1}{2}})\biggr{\}}.

By (6.2), we are done. The second claim is obvious from the formula of 𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a).

10.1.3. The proof of Theorem 6.4 (3)

Let vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}). From (3.10), we have the equality

𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎\displaystyle{\mathfrak{F}}_{v}^{(z)}(a) =Fv{𝕂0(𝔭)Φv(k1[a(a1)x01]k)dk\displaystyle=\int_{F_{v}}\{\int_{{\mathbb{K}}_{0}({\mathfrak{p}})}\Phi_{v}\left(k^{-1}\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]k\right){{d}}k
+ξ𝔬/𝔭𝕂0(𝔭)Φv(k1w01[1ξ01][a(a1)x01][1ξ01]w0k)dk}φv(0,z)([1x01])dx,\displaystyle+\sum_{\xi\in\mathfrak{o}/{\mathfrak{p}}}\int_{{\mathbb{K}}_{0}({\mathfrak{p}})}\Phi_{v}\left(k^{-1}w_{0}^{-1}\left[\begin{smallmatrix}1&-\xi\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}1&\xi\\ 0&1\end{smallmatrix}\right]w_{0}k\right)\,{{d}}k\}\,\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x,

from which we see that 𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a) is vol(𝕂0(𝔭v))volsubscript𝕂0subscript𝔭𝑣{\operatorname{vol}}({\mathbb{K}}_{0}({\mathfrak{p}}_{v})) times the sum of the following integrals

(10.7) FvΦv([a(a1)x01])φv(0,z)([1x01])𝑑x,subscriptsubscript𝐹𝑣subscriptΦ𝑣delimited-[]𝑎𝑎1𝑥01superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥\displaystyle\textstyle{\int}_{F_{v}}\Phi_{v}\left(\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]\right)\,\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x,\
(10.8) FvΦv([10(a1)(x+ξ)a])φv(0,z)([1x01])𝑑xfor ξ𝔬/𝔭,subscriptsubscript𝐹𝑣subscriptΦ𝑣delimited-[]10𝑎1𝑥𝜉𝑎superscriptsubscript𝜑𝑣0𝑧delimited-[]1𝑥01differential-d𝑥for ξ𝔬/𝔭\displaystyle\textstyle{\int}_{F_{v}}\Phi_{v}\left(\left[\begin{smallmatrix}1&0\\ -(a-1)(x+\xi)&a\end{smallmatrix}\right]\right)\,\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x\quad\text{for $\xi\in\mathfrak{o}/{\mathfrak{p}}$},

where Φv=chZv𝕂0(𝔭)subscriptΦ𝑣subscriptchsubscript𝑍𝑣subscript𝕂0𝔭\Phi_{v}={\rm ch}_{Z_{v}{\mathbb{K}}_{0}({\mathfrak{p}})}. Since Φv([a(a1)x01])1subscriptΦ𝑣delimited-[]𝑎𝑎1𝑥011\Phi_{v}\left(\left[\begin{smallmatrix}a&(a-1)x\\ 0&1\end{smallmatrix}\right]\right)\not=1 if and only if x(a1)1𝔬𝑥superscript𝑎11𝔬x\in(a-1)^{-1}\mathfrak{o}, a𝔬×𝑎superscript𝔬a\in\mathfrak{o}^{\times}, from the proof of Theorem 6.4 (2) in §10.1.2, the integral (10.7) equals δ(a𝔬×)qd/2𝒪0,v1,(z)((a1)2)𝛿𝑎superscript𝔬superscript𝑞𝑑2superscriptsubscript𝒪0𝑣1𝑧superscript𝑎12\delta(a\in\mathfrak{o}^{\times})q^{-d/2}{\mathcal{O}}_{0,v}^{1,(z)}((a-1)^{-2}). The integral (10.8) is computed as δ(a𝔬×){Av(0,z)Iξ(z)+Av(0,z)Iξ(z)}𝛿𝑎superscript𝔬subscript𝐴𝑣0𝑧subscript𝐼𝜉𝑧subscript𝐴𝑣0𝑧subscript𝐼𝜉𝑧\delta(a\in\mathfrak{o}^{\times})\{A_{v}(0,z)I_{\xi}(z)+A_{v}(0,-z)I_{\xi}(-z)\}, where Iξ(z)=xξ+(a1)1𝔭hv(0,z)(x)𝑑xsubscript𝐼𝜉𝑧subscript𝑥𝜉superscript𝑎11𝔭superscriptsubscript𝑣0𝑧𝑥differential-d𝑥I_{\xi}(z)=\int_{x\in-\xi+(a-1)^{-1}{\mathfrak{p}}}h_{v}^{(0,z)}(x){{d}}x. Let a𝔬×𝑎superscript𝔬a\in\mathfrak{o}^{\times}. Suppose |a1|=1𝑎11|a-1|=1. Then we easily have Iξ(z)=vol(𝔭)=q1d/2subscript𝐼𝜉𝑧vol𝔭superscript𝑞1𝑑2I_{\xi}(z)={\operatorname{vol}}({\mathfrak{p}})=q^{-1-d/2}, and (10.8) becomes q1d/2(Av(0,z)+Av(0,z))=q1d/2superscript𝑞1𝑑2subscript𝐴𝑣0𝑧subscript𝐴𝑣0𝑧superscript𝑞1𝑑2q^{-1-d/2}(A_{v}(0,z)+A_{v}(0,-z))=q^{-1-d/2}. Hence 𝔉v(z)(a)=vol(𝕂0(𝔭))(qd/2+q×q1d/2)=2qd/2(1+q)1.superscriptsubscript𝔉𝑣𝑧𝑎volsubscript𝕂0𝔭superscript𝑞𝑑2𝑞superscript𝑞1𝑑22superscript𝑞𝑑2superscript1𝑞1{\mathfrak{F}}_{v}^{(z)}(a)={\operatorname{vol}}({\mathbb{K}}_{0}({\mathfrak{p}}))\,(q^{-d/2}+q\times q^{-1-d/2})={2q^{-d/2}}{(1+q)^{-1}}. Suppose |a1|<1𝑎11|a-1|<1. Noting ξ𝔬(a1)1𝔭𝜉𝔬superscript𝑎11𝔭-\xi\in\mathfrak{o}\subset(a-1)^{-1}{\mathfrak{p}}, we have

Iξ(z)subscript𝐼𝜉𝑧\displaystyle I_{\xi}(z) =x(a1)1𝔭hv(0,z)(x)𝑑x=I(z)hv(0,z)((a1)1)qd/2(1q1)|a1|1,absentsubscript𝑥superscript𝑎11𝔭superscriptsubscript𝑣0𝑧𝑥differential-d𝑥𝐼𝑧superscriptsubscript𝑣0𝑧superscript𝑎11superscript𝑞𝑑21superscript𝑞1superscript𝑎11\displaystyle=\textstyle{\int}_{x\in(a-1)^{-1}{\mathfrak{p}}}h_{v}^{(0,z)}(x)\,{{d}}x=I(z)-h_{v}^{(0,z)}((a-1)^{-1})\,q^{-d/2}(1-q^{-1})|a-1|^{-1},

where I(z)𝐼𝑧I(z) is the same integral as in the proof of Theorem 6.4 (2) in §10.1.2. Thus (10.8) is the sum of Av(0,z)I(z)+Av(0,z)I(z)subscript𝐴𝑣0𝑧𝐼𝑧subscript𝐴𝑣0𝑧𝐼𝑧A_{v}(0,z)I(z)+A_{v}(0,-z)I(-z), which becomes δ(a𝔬×)qd/2𝒪0,v1,(z)((a1)2)𝛿𝑎superscript𝔬superscript𝑞𝑑2superscriptsubscript𝒪0𝑣1𝑧superscript𝑎12\delta(a\in\mathfrak{o}^{\times})q^{-d/2}{\mathcal{O}}_{0,v}^{1,(z)}((a-1)^{-2}) as before and qd/2(1q1){Av(0,z)|a1|z12+Av(0,z)|a1|z12}.superscript𝑞𝑑21superscript𝑞1subscript𝐴𝑣0𝑧superscript𝑎1𝑧12subscript𝐴𝑣0𝑧superscript𝑎1𝑧12-q^{-d/2}(1-q^{-1})\,\{A_{v}(0,z)|a-1|^{\frac{z-1}{2}}+A_{v}(0,-z)|a-1|^{\frac{-z-1}{2}}\}. Hence 𝔉v(z)(a)superscriptsubscript𝔉𝑣𝑧𝑎{\mathfrak{F}}_{v}^{(z)}(a) equals the following expression multiplied by vol(𝕂0(𝔭))qd/2volsubscript𝕂0𝔭superscript𝑞𝑑2{\operatorname{vol}}({\mathbb{K}}_{0}({\mathfrak{p}}))q^{-d/2}:

(1+q)𝒪0,v1,(z)((a1)2)q×(1qv1)(Av(0,z)|a1|z12+Av(0,z)|a1|z12).1𝑞superscriptsubscript𝒪0𝑣1𝑧superscript𝑎12𝑞1superscriptsubscript𝑞𝑣1subscript𝐴𝑣0𝑧superscript𝑎1𝑧12subscript𝐴𝑣0𝑧superscript𝑎1𝑧12\displaystyle(1+q){\mathcal{O}}_{0,v}^{1,(z)}((a-1)^{-2})-q\times(1-q_{v}^{-1})\,\left(A_{v}(0,z)|a-1|^{\frac{z-1}{2}}+A_{v}(0,-z)|a-1|^{\frac{-z-1}{2}}\right).

From this we get the desired formula by a short computation.

10.1.4. The proof of Theorem 6.4 (4)

Let vS𝑣𝑆v\in S. Put s=sv𝑠subscript𝑠𝑣s=s_{v}. By applying (2.3) and (6.1) and then changing the order of integrals,

(10.9) 𝔉v(z)(s;a)superscriptsubscript𝔉𝑣𝑧𝑠𝑎\displaystyle{\mathfrak{F}}_{v}^{(z)}(s;a) =|a1|1FvΦ^v(s;[ax01])φv(0,z)([1(a1)1x01])𝑑xabsentsuperscript𝑎11subscriptsubscript𝐹𝑣subscript^Φ𝑣𝑠delimited-[]𝑎𝑥01superscriptsubscript𝜑𝑣0𝑧delimited-[]1superscript𝑎11𝑥01differential-d𝑥\displaystyle=|a-1|^{-1}\textstyle{\int}_{F_{v}}\hat{\Phi}_{v}\left(s;\left[\begin{smallmatrix}a&x\\ 0&1\end{smallmatrix}\right]\right)\,\varphi_{v}^{(0,z)}\left(\left[\begin{smallmatrix}1&(a-1)^{-1}x\\ 0&1\end{smallmatrix}\right]\right)\,{{d}}x
=(qs+12)|a1|1|a|(s+1)/2ζFv(s+1){Av(0,z)I(z,s;a)+Av(0,z)I(z,s;a)},absentsuperscript𝑞𝑠12superscript𝑎11superscript𝑎𝑠12subscript𝜁subscript𝐹𝑣𝑠1subscript𝐴𝑣0𝑧𝐼𝑧𝑠𝑎subscript𝐴𝑣0𝑧𝐼𝑧𝑠𝑎\displaystyle=(-q^{-\frac{s+1}{2}})|a-1|^{-1}{|a|^{(s+1)/2}\zeta_{F_{v}}(s+1)}\,\{A_{v}(0,z)I(z,s;a)+A_{v}(0,-z)I(-z,s;a)\},

where we set

I(z,s;a)=Fvmax(1,|a|,|x|)(s+1)max(1,|a1|1|x|)z+12dx.\displaystyle I(z,s;a)=\textstyle{\int}_{F_{v}}\max(1,|a|,|x|)^{-(s+1)}\,\max(1,|a-1|^{-1}|x|)^{-\frac{z+1}{2}}\,{{d}}x.

As will be shown below, the integral I(±z,s;a)𝐼plus-or-minus𝑧𝑠𝑎I(\pm z,s;a) is absolutely convergent if Re(s)>(|Re(z)|1)/2Re𝑠Re𝑧12\operatorname{Re}(s)>(|\operatorname{Re}(z)|-1)/2. Suppose |a1|<1𝑎11|a-1|<1. Then |a|=1𝑎1|a|=1. Hence

I(z,s;a)𝐼𝑧𝑠𝑎\displaystyle I(z,s;a) =|x||a1|𝑑x+|a1|<|x|1(|a1|1|x|)z+12𝑑x+1<|x||x|(s+1)(|a1|1|x|)z+12𝑑xabsentsubscript𝑥𝑎1differential-d𝑥subscript𝑎1𝑥1superscriptsuperscript𝑎11𝑥𝑧12differential-d𝑥subscript1𝑥superscript𝑥𝑠1superscriptsuperscript𝑎11𝑥𝑧12differential-d𝑥\displaystyle=\textstyle{\int}_{|x|\leqslant|a-1|}\,{{d}}x+\textstyle{\int}_{|a-1|<|x|\leqslant 1}(|a-1|^{-1}|x|)^{-\frac{z+1}{2}}\,{{d}}x+\textstyle{\int}_{1<|x|}|x|^{-(s+1)}(\,|a-1|^{-1}|x|)^{-\frac{z+1}{2}}\,{{d}}x
=qd/2|a1|+|a1|z+12qd/2(1q1)(1|a1|z12)(1qz12)1absentsuperscript𝑞𝑑2𝑎1superscript𝑎1𝑧12superscript𝑞𝑑21superscript𝑞11superscript𝑎1𝑧12superscript1superscript𝑞𝑧121\displaystyle=q^{-d/2}|a-1|+|a-1|^{\frac{z+1}{2}}\,q^{-d/2}(1-q^{-1}){(1-|a-1|^{-\frac{z-1}{2}})}{(1-q^{\frac{z-1}{2}})^{-1}}
+|a1|z+12qd/2(1q1)qz12s1(1qz12s1)1superscript𝑎1𝑧12superscript𝑞𝑑21superscript𝑞1superscript𝑞𝑧12𝑠1superscript1superscript𝑞𝑧12𝑠11\displaystyle\qquad+|a-1|^{\frac{z+1}{2}}\,q^{-d/2}(1-q^{-1})\,{q^{-\frac{z-1}{2}-s-1}}{(1-q^{-\frac{z-1}{2}-s-1})^{-1}}

if Re(z12+s+1)>0Re𝑧12𝑠10\operatorname{Re}(\frac{z-1}{2}+s+1)>0. By a computation, we obtain

I(z,s;a)=qd/2|a1|{qz12ζFv(1z2)ζFv(1+z2)+ζFv(1z2)ζFv(z12+s+1)ζFv(1)ζFv(s+1)|a1|z12}.𝐼𝑧𝑠𝑎superscript𝑞𝑑2𝑎1superscript𝑞𝑧12subscript𝜁subscript𝐹𝑣1𝑧2subscript𝜁subscript𝐹𝑣1𝑧2subscript𝜁subscript𝐹𝑣1𝑧2subscript𝜁subscript𝐹𝑣𝑧12𝑠1subscript𝜁subscript𝐹𝑣1subscript𝜁subscript𝐹𝑣𝑠1superscript𝑎1𝑧12\displaystyle I(z,s;a)=q^{-d/2}|a-1|\,\left\{-q^{\frac{z-1}{2}}\frac{\zeta_{F_{v}}\left(\frac{1-z}{2}\right)}{\zeta_{F_{v}}\left(\frac{1+z}{2}\right)}+\frac{\zeta_{F_{v}}\left(\frac{1-z}{2}\right)\zeta_{F_{v}}\left(\frac{z-1}{2}+s+1\right)}{\zeta_{F_{v}}(1)\zeta_{F_{v}}(s+1)}|a-1|^{\frac{z-1}{2}}\right\}.

From this, combined with (6.3), we get the formula

(10.10) Av(0,z)I(z,s;a)+Av(0,z)I(z,s;a)subscript𝐴𝑣0𝑧𝐼𝑧𝑠𝑎subscript𝐴𝑣0𝑧𝐼𝑧𝑠𝑎\displaystyle A_{v}(0,z)I(z,s;a)+A_{v}(0,-z)I(-z,s;a) =(qs+12)|a1|ζFv(s+1)1qd/2𝒮v1,(z)((a1)2).absentsuperscript𝑞𝑠12𝑎1subscript𝜁subscript𝐹𝑣superscript𝑠11superscript𝑞𝑑2superscriptsubscript𝒮𝑣1𝑧superscript𝑎12\displaystyle=(-q^{\frac{s+1}{2}}){|a-1|}{\zeta_{F_{v}}(s+1)^{-1}}\,q^{-d/2}{\mathcal{S}}_{v}^{1,(z)}((a-1)^{-2}).

Suppose |a1|>1𝑎11|a-1|>1. Then |a1|=|a|𝑎1𝑎|a-1|=|a| and

I(z,s;a)𝐼𝑧𝑠𝑎\displaystyle I(z,s;a) =|x||a1|max(1,|x|,|a|)(s+1)dx+|a1|<|x||x|(s+1)(|a1|1|x|)z+12dx\displaystyle=\textstyle{\int}_{|x|\leqslant|a-1|}\max(1,|x|,|a|)^{-(s+1)}\,{{d}}x+\textstyle{\int}_{|a-1|<|x|}|x|^{-(s+1)}(|a-1|^{-1}|x|)^{-\frac{z+1}{2}}\,{{d}}x
=qd/2|a1|s(1qz+12s1)(1qz12s1)1absentsuperscript𝑞𝑑2superscript𝑎1𝑠1superscript𝑞𝑧12𝑠1superscript1superscript𝑞𝑧12𝑠11\displaystyle=q^{-d/2}|a-1|^{-s}\,{(1-q^{-\frac{z+1}{2}-s-1})}{(1-q^{-\frac{z-1}{2}-s-1})^{-1}}

if Re(z12+s+1)>0Re𝑧12𝑠10\operatorname{Re}(\tfrac{z-1}{2}+s+1)>0. Suppose |a1|=1𝑎11|a-1|=1. Then |a|1𝑎1|a|\leqslant 1 and

I(z,s;a)𝐼𝑧𝑠𝑎\displaystyle I(z,s;a) =|x|1dx+1<|x|max(|a|,|x|)(s+1)|x|z+12dx\displaystyle=\textstyle{\int}_{|x|\leqslant 1}{{d}}x+\int_{1<|x|}\max(|a|,|x|)^{-(s+1)}|x|^{-\frac{z+1}{2}}\,{{d}}x
=qd/2{1+(1q1)qz12s1(1qz12s1)1}absentsuperscript𝑞𝑑211superscript𝑞1superscript𝑞𝑧12𝑠1superscript1superscript𝑞𝑧12𝑠11\displaystyle=q^{-d/2}\left\{1+{(1-q^{-1})\,q^{-\frac{z-1}{2}-s-1}}{(1-q^{-\frac{z-1}{2}-s-1})^{-1}}\right\}
=qd/2(1qz+12s1)(1qz12s1)1absentsuperscript𝑞𝑑21superscript𝑞𝑧12𝑠1superscript1superscript𝑞𝑧12𝑠11\displaystyle=q^{-d/2}{(1-q^{-\frac{z+1}{2}-s-1})}{(1-q^{-\frac{z-1}{2}-s-1})^{-1}}

if Re(z12+s+1)>0Re𝑧12𝑠10\operatorname{Re}(\tfrac{z-1}{2}+s+1)>0. Hence I(z,s;a)=qd/2|a1|sζFv(z12+s+1)ζFv(z+12+s+1)1𝐼𝑧𝑠𝑎superscript𝑞𝑑2superscript𝑎1𝑠subscript𝜁subscript𝐹𝑣𝑧12𝑠1subscript𝜁subscript𝐹𝑣superscript𝑧12𝑠11I(z,s;a)=q^{-d/2}|a-1|^{-s}{\zeta_{F_{v}}\left(\frac{z-1}{2}+s+1\right)}{\zeta_{F_{v}}\left(\frac{z+1}{2}+s+1\right)^{-1}} for |a1|1𝑎11|a-1|\geqslant 1. From this, by a direct computation, we get the formula (10.10) again.

10.2. Local elliptic orbital integrals

Let vΣF𝑣subscriptΣ𝐹v\in\Sigma_{F}. In this subsection, we compute the integral (7.9) with Δv01superscriptsubscriptΔ𝑣01\Delta_{v}^{0}\not=1 to complete the proof of Theorem 7.9. In this section throughout, we fix (t:n)F𝒬FIrr(t:n)_{F}\in{\mathcal{Q}}_{F}^{{\rm Irr}} with the decomposition (t24n)(v)=Δv0(2mv)2superscriptsuperscript𝑡24𝑛𝑣superscriptsubscriptΔ𝑣0superscript2subscript𝑚𝑣2(t^{2}-4n)^{(v)}=\Delta_{v}^{0}(2m_{v})^{2} at a place v𝑣v as before. Set a=t2mv𝑎𝑡2subscript𝑚𝑣a=\tfrac{t}{2m_{v}} and τ=Δv0𝜏superscriptsubscriptΔ𝑣0\tau=\Delta_{v}^{0} to simplify notation. Recall the construction at the begining of § 7.2. By multiplying φ0,v(12)subscript𝜑0𝑣subscript12\varphi_{0,v}(1_{2}), the integral 𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) is transformed to

(10.11) φ0,v(12)𝔈v(z)(γ^v)=Zv\GvΦv(g1[a1τa]g)f0,v(g)𝑑gsubscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣subscript\subscript𝑍𝑣subscript𝐺𝑣subscriptΦ𝑣superscript𝑔1delimited-[]𝑎1𝜏𝑎𝑔subscript𝑓0𝑣𝑔differential-d𝑔\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=\textstyle{\int}_{Z_{v}\backslash G_{v}}\Phi_{v}\left(g^{-1}\left[\begin{smallmatrix}a&1\\ \tau&a\end{smallmatrix}\right]g\right)\,f_{0,v}(g)\,{{d}}g
=\displaystyle= Fv×Fv𝕂v𝑑kΦv(k1[t001]1[1x01][a1τa][1x01][t001]k)f0,v([1x01][t001])|t|v1d×t𝑑xsubscriptsuperscriptsubscript𝐹𝑣subscriptsubscript𝐹𝑣subscriptsubscript𝕂𝑣differential-d𝑘subscriptΦ𝑣superscript𝑘1superscriptdelimited-[]𝑡0011delimited-[]1𝑥01delimited-[]𝑎1𝜏𝑎delimited-[]1𝑥01delimited-[]𝑡001𝑘subscript𝑓0𝑣delimited-[]1𝑥01delimited-[]𝑡001superscriptsubscript𝑡𝑣1superscript𝑑𝑡differential-d𝑥\displaystyle\textstyle{\int}_{F_{v}^{\times}}\int_{F_{v}}\int_{{\mathbb{K}}_{v}}{{d}}k\,\Phi_{v}\left(k^{-1}\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]^{-1}\left[\begin{smallmatrix}1&-x\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}a&1\\ \tau&a\end{smallmatrix}\right]\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]k\right)\,f_{0,v}\left(\left[\begin{smallmatrix}1&x\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}t&0\\ 0&1\end{smallmatrix}\right]\right)\,|t|_{v}^{-1}{{d}}^{\times}t\,{{d}}x
=\displaystyle= Fv×Fv𝕂v𝑑kΦv(k1[aτxt1(1τx2)τta+τx]k)|t|vz12d×t𝑑x.subscriptsuperscriptsubscript𝐹𝑣subscriptsubscript𝐹𝑣subscriptsubscript𝕂𝑣differential-d𝑘subscriptΦ𝑣superscript𝑘1delimited-[]𝑎𝜏𝑥superscript𝑡11𝜏superscript𝑥2𝜏𝑡𝑎𝜏𝑥𝑘superscriptsubscript𝑡𝑣𝑧12superscript𝑑𝑡differential-d𝑥\displaystyle\textstyle{\int}_{F_{v}^{\times}}\textstyle{\int}_{F_{v}}\textstyle{\int}_{{\mathbb{K}}_{v}}{{d}}k\,\Phi_{v}\left(k^{-1}\left[\begin{smallmatrix}a-\tau x&t^{-1}(1-\tau x^{2})\\ \tau t&a+\tau x\end{smallmatrix}\right]k\right)\,|t|_{v}^{\frac{z-1}{2}}{{d}}^{\times}t\,{{d}}x.

10.2.1. The proof of Theorem 7.9 (1)

Let vΣ𝑣subscriptΣv\in\Sigma_{\infty} and Δv0=1superscriptsubscriptΔ𝑣01\Delta_{v}^{0}=-1. By 𝐊v=𝐊v0[1001]𝐊v0subscript𝐊𝑣superscriptsubscript𝐊𝑣0delimited-[]1001superscriptsubscript𝐊𝑣0{\mathbf{K}}_{v}={\mathbf{K}}_{v}^{0}\cup[\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix}]{\mathbf{K}}_{v}^{0} and by (b) in § 2.2, we have

φ0,v(12)𝔈v(z)(γ^v)=Fv×B(t)|t|z12d×tsubscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡superscript𝑡𝑧12superscript𝑑𝑡\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=\textstyle{\int}_{F_{v}^{\times}}B(t)|t|^{\frac{z-1}{2}}\,d^{\times}t

with

B(t)=FvΦl(a+xt1(1+x2)tax)𝑑x=2l(1+a2)l/2Fv(2ai(t1x2+t1+t))l𝑑x.𝐵𝑡subscriptsubscript𝐹𝑣superscriptΦ𝑙𝑎𝑥superscript𝑡11superscript𝑥2𝑡𝑎𝑥differential-d𝑥superscript2𝑙superscript1superscript𝑎2𝑙2subscriptsubscript𝐹𝑣superscript2𝑎𝑖superscript𝑡1superscript𝑥2superscript𝑡1𝑡𝑙differential-d𝑥B(t)=\textstyle{\int}_{F_{v}}\Phi^{l}(\begin{smallmatrix}a+x&t^{-1}(1+x^{2})\\ -t&a-x\end{smallmatrix})dx={2^{l}(1+a^{2})^{l/2}}\textstyle{\int}_{F_{v}}\left(2a-i(t^{-1}x^{2}+t^{-1}+t)\right)^{-l}dx.

By the formula (1+zx2)l𝑑x=Γ(l1/2)πΓ(l)1z1subscriptsuperscript1𝑧superscript𝑥2𝑙differential-d𝑥Γ𝑙12𝜋Γsuperscript𝑙1superscript𝑧1\textstyle{\int}_{\mathbb{R}}{(1+zx^{2})^{-l}}dx={\Gamma(l-1/2)\sqrt{\pi}}{\Gamma(l)^{-1}}\,\sqrt{z}^{-1} (Re(z)>0)\operatorname{Re}(z)>0) easily proved by the residue theorem,

B(t)=2l(1+a2)l/2Γ(l1/2)πΓ(l)1×(it)l(t2+1+2ati)1/2l,𝐵𝑡superscript2𝑙superscript1superscript𝑎2𝑙2Γ𝑙12𝜋Γsuperscript𝑙1superscript𝑖𝑡𝑙superscriptsuperscript𝑡212𝑎𝑡𝑖12𝑙\displaystyle B(t)=2^{l}(1+a^{2})^{l/2}{\Gamma(l-1/2)\sqrt{\pi}}{\Gamma(l)^{-1}}\times(it)^{l}(t^{2}+1+2ati)^{1/2-l},

Since l𝑙l is even, by decomposing the t𝑡t-integral over +superscript{\mathbb{R}}^{+} into positive and negative reals, we obtain

Fv×B(t)|t|z12d×t=2l(1+a2)l/2il{Il(4,2a;z+12)+Il(4,2a;z+12)},subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡superscript𝑡𝑧12superscript𝑑𝑡superscript2𝑙superscript1superscript𝑎2𝑙2superscript𝑖𝑙subscript𝐼𝑙42𝑎𝑧12subscript𝐼𝑙42𝑎𝑧12\displaystyle\textstyle{\int}_{F_{v}^{\times}}B(t)|t|^{\frac{z-1}{2}}d^{\times}t=2^{l}(1+a^{2})^{l/2}\,i^{l}\,\{I_{l}\left(-4,2a;\tfrac{z+1}{2}\right)+I_{l}\left(-4,-2a;\tfrac{z+1}{2}\right)\},

where Il(Δ,a;s)subscript𝐼𝑙Δ𝑎𝑠I_{l}(\Delta,a;s) is the Zagier’s function defined in [35, p.110]. We have

Il(4,2a;z+12)subscript𝐼𝑙42𝑎𝑧12\displaystyle I_{l}\left(-4,2a;\tfrac{z+1}{2}\right) =Γ(l1/2)πΓ(l)10yl+z12(y2+2iay+1)l1/2d×yabsentΓ𝑙12𝜋Γsuperscript𝑙1superscriptsubscript0superscript𝑦𝑙𝑧12superscriptsuperscript𝑦22𝑖𝑎𝑦1𝑙12superscript𝑑𝑦\displaystyle=\Gamma(l-1/2)\sqrt{\pi}\Gamma(l)^{-1}\textstyle{\int}_{0}^{\infty}\tfrac{y^{l+\frac{z-1}{2}}}{(y^{2}+2iay+1)^{l-1/2}}{{d}}^{\times}y
=21lπΓ(l)1Γ(l+z12)Γ(l+z12)(a21)l12𝔓z121l(ai).absentsuperscript21𝑙𝜋Γsuperscript𝑙1Γ𝑙𝑧12Γ𝑙𝑧12superscriptsuperscript𝑎21𝑙12superscriptsubscript𝔓𝑧121𝑙𝑎𝑖\displaystyle=2^{1-l}\pi\Gamma(l)^{-1}{\Gamma\left(l+\tfrac{z-1}{2}\right)\Gamma\left(l+\tfrac{-z-1}{2}\right)}{(-a^{2}-1)^{-\frac{l-1}{2}}}{\mathfrak{P}}_{\frac{z-1}{2}}^{1-l}(ai).

for |Re(z)|<2l1Re𝑧2𝑙1|\operatorname{Re}(z)|<2l-1 by means of the formula [9, p.961, 8.713, 3], where the square root of ai±1plus-or-minus𝑎𝑖1ai\pm 1 is chosen so that arg(ai±1)(π,π)plus-or-minus𝑎𝑖1𝜋𝜋\arg(ai\pm 1)\in(-\pi,\pi). Consequently, by noting 1a2=sgn(a)ia2+11superscript𝑎2sgn𝑎𝑖superscript𝑎21\sqrt{-1-a^{2}}={\operatorname{sgn}}(a)i\sqrt{a^{2}+1}, we have

φ0,v(12)𝔈v(z)(γ^v)=subscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣absent\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})= sgn(a)i1+a2×2πΓ(l+z12)Γ(l+z12)Γ(l)1(𝔓z121l(ai)𝔓z121l(ai)).sgn𝑎𝑖1superscript𝑎22𝜋Γ𝑙𝑧12Γ𝑙𝑧12Γsuperscript𝑙1superscriptsubscript𝔓𝑧121𝑙𝑎𝑖superscriptsubscript𝔓𝑧121𝑙𝑎𝑖\displaystyle{\operatorname{sgn}}(a)i\sqrt{1+a^{2}}\times 2\pi{\Gamma\left(l+\tfrac{z-1}{2}\right)\Gamma\left(l+\tfrac{-z-1}{2}\right)}{\Gamma(l)^{-1}}\left({\mathfrak{P}}_{\frac{z-1}{2}}^{1-l}(ai)-{\mathfrak{P}}_{\frac{z-1}{2}}^{1-l}(-ai)\right).

We complete the proof by Lemma 7.4.

10.2.2. The proof of Theorem 7.9 (2) and (3)

Since Φv=chZv𝕂vsubscriptΦ𝑣subscriptchsubscript𝑍𝑣subscript𝕂𝑣\Phi_{v}={\rm ch}_{Z_{v}{\mathbb{K}}_{v}}, the integral domain of (10.11) is restricted to only those (t,x)Fv××Fv𝑡𝑥superscriptsubscript𝐹𝑣subscript𝐹𝑣(t,x)\in F_{v}^{\times}\times F_{v} such that

(10.12) c1(aτx)𝔬,c1(a+τx)𝔬,c1t1(1τx2)𝔬,formulae-sequencesuperscript𝑐1𝑎𝜏𝑥𝔬formulae-sequencesuperscript𝑐1𝑎𝜏𝑥𝔬superscript𝑐1superscript𝑡11𝜏superscript𝑥2𝔬\displaystyle c^{-1}(a-\tau x)\in\mathfrak{o},\quad c^{-1}(a+\tau x)\in\mathfrak{o},\quad c^{-1}t^{-1}(1-\tau x^{2})\in\mathfrak{o},
(10.13) c1τt𝔬,superscript𝑐1𝜏𝑡𝔬\displaystyle c^{-1}\tau t\in\mathfrak{o},
(10.14) c2(a2τ)𝔬×superscript𝑐2superscript𝑎2𝜏superscript𝔬\displaystyle c^{-2}(a^{2}-\tau)\in\mathfrak{o}^{\times}

with some cFv×𝑐superscriptsubscript𝐹𝑣c\in F_{v}^{\times}.

(I) The case |a||τ|𝑎𝜏|a|\leqslant|\tau|.

(i) Suppose vΣdyadic𝑣subscriptΣdyadicv\not\in\Sigma_{\rm dyadic}, τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in\mathfrak{o}^{\times}-(\mathfrak{o}^{\times})^{2}. From Lemma 7.1, a2τ𝔬×superscript𝑎2𝜏superscript𝔬a^{2}-\tau\in\mathfrak{o}^{\times}. Hence from (10.14) we get c𝔬×𝑐superscript𝔬c\in\mathfrak{o}^{\times}, which, combined with (10.13), yields t𝔬𝑡𝔬t\in\mathfrak{o}. From the last relation in (10.12), we have the containment 1τx2t𝔬1𝜏superscript𝑥2𝑡𝔬1-\tau x^{2}\in t\mathfrak{o} from which x𝔬𝑥𝔬x\in\mathfrak{o} follows. If t𝑡t were non-unit, then 1τx2t𝔬𝔭1𝜏superscript𝑥2𝑡𝔬𝔭1-\tau x^{2}\in t\mathfrak{o}\subset{\mathfrak{p}}, which is impossible due to Lemma 7.1. Hence we have that the existence of c𝑐c satisfying (10.12), (10.13) and (10.14) is equivalent to t𝔬×𝑡superscript𝔬t\in\mathfrak{o}^{\times} and x𝔬𝑥𝔬x\in\mathfrak{o}. Thus by Lemm 7.4, we have 𝔈v(z)(γ^v)=φ0,v(12)1𝔬×d×t𝔬𝑑x=|mv|qd/2superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣subscript𝜑0𝑣superscriptsubscript121subscriptsuperscript𝔬superscript𝑑𝑡subscript𝔬differential-d𝑥subscript𝑚𝑣superscript𝑞𝑑2{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=\varphi_{0,v}(1_{2})^{-1}\textstyle{\int}_{\mathfrak{o}^{\times}}{{d}}^{\times}t\textstyle{\int}_{\mathfrak{o}}{{d}}x=|m_{v}|q^{-d/2}.

(ii) Suppose vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic}, τ{5,1,5}𝜏515\tau\in\{-5,-1,5\}. Let us consider the case τ=5𝜏5\tau=5. If |a|=1𝑎1|a|=1, then (10.14) implies c22×𝑐2superscriptsubscript2c\in 2\mathbb{Z}_{2}^{\times}. Combining this with (10.12) and (10.13), we obtain x2×𝑥superscriptsubscript2x\in\mathbb{Z}_{2}^{\times} and t22×𝑡2superscriptsubscript2t\in 2\mathbb{Z}_{2}^{\times}. Hence by Lemma 7.4,

𝔈v(z)(γ^v)=φv,0(12)1|t|=|2||x|=1|t|(z1)/2d×t𝑑x=32|mv| 2d/22(z1)/21+2z.superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣subscript𝜑𝑣0superscriptsubscript121subscript𝑡2subscript𝑥1superscript𝑡𝑧12superscript𝑑𝑡differential-d𝑥32subscript𝑚𝑣superscript2𝑑2superscript2𝑧121superscript2𝑧{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=\varphi_{v,0}(1_{2})^{-1}\textstyle{\int}_{|t|=|2|}\textstyle{\int}_{|x|=1}|t|^{(z-1)/2}d^{\times}tdx=\tfrac{3}{2}|m_{v}|\,2^{-d/2}\tfrac{2^{(-z-1)/2}}{1+2^{-z}}.

If |a|<1𝑎1|a|<1, the condition (10.14) implies c2×𝑐subscriptsuperscript2c\in{\mathbb{Z}}^{\times}_{2}. Then using (10.12) and (10.13), we get x2𝑥subscript2x\in{\mathbb{Z}}_{2} and 15x2t215superscript𝑥2𝑡subscript21-5x^{2}\in t{\mathbb{Z}}_{2}. Thus |4||t|14𝑡1|4|\leqslant|t|\leqslant 1 if |x|=1𝑥1|x|=1 and |t|=1𝑡1|t|=1 if |x|<1𝑥1|x|<1, and whence

𝔈v(z)(γ^v)=superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣absent\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})= φ0,v(12)1(|4||t|1d×t|x|=1𝑑x|t|(z1)/2+|t|=1d×t|x|<1𝑑x|t|(z1)/2)subscript𝜑0𝑣superscriptsubscript121subscript4𝑡1superscript𝑑𝑡subscript𝑥1differential-d𝑥superscript𝑡𝑧12subscript𝑡1superscript𝑑𝑡subscript𝑥1differential-d𝑥superscript𝑡𝑧12\displaystyle\varphi_{0,v}(1_{2})^{-1}\bigg{(}\textstyle{\int}_{|4|\leqslant|t|\leqslant 1}d^{\times}t\textstyle{\int}_{|x|=1}dx|t|^{(z-1)/2}+\textstyle{\int}_{|t|=1}d^{\times}t\textstyle{\int}_{|x|<1}dx|t|^{(z-1)/2}\bigg{)}
=\displaystyle= 32|mv|v2d/2{1+2(z1)/2+2z}/(1+2z).32subscriptsubscript𝑚𝑣𝑣superscript2𝑑21superscript2𝑧12superscript2𝑧1superscript2𝑧\displaystyle\tfrac{3}{2}|m_{v}|_{v}2^{-d/2}\{1+2^{(-z-1)/2}+2^{-z}\}/(1+2^{-z}).

We note |nmv2|=1𝑛superscriptsubscript𝑚𝑣21|\frac{n}{m_{v}^{2}}|=1 from Lemma 7.11 (4) and Proposition 7.12 (1). Next consider the case τ{5,1}𝜏51\tau\in\{-5,-1\}. If |a|=1𝑎1|a|=1, then a2τ22×superscript𝑎2𝜏2superscriptsubscript2a^{2}-\tau\in 2{\mathbb{Z}}_{2}^{\times}, which contradicts to (10.14). Thus 𝔈v(z)(γ^v)=0superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣0{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=0. If |a|<1𝑎1|a|<1, then (10.14) implies c2×𝑐superscriptsubscript2c\in\mathbb{Z}_{2}^{\times}. We may set c=1𝑐1c=1. Then, x2𝑥subscript2x\in\mathbb{Z}_{2} and 1τx2t21𝜏superscript𝑥2𝑡subscript21-\tau x^{2}\in t\mathbb{Z}_{2}. Thus, |2||t|12𝑡1|2|\leqslant|t|\leqslant 1 if |x|=1𝑥1|x|=1 and |t|=1𝑡1|t|=1 if |x|<1𝑥1|x|<1. We have

𝔈v(z)(γ^v)=superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣absent\displaystyle{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})= φ0,v(12)1(|2||t|1d×t|x|=1𝑑x|t|(z1)/2+|t|=1d×t|x|<1𝑑x|t|(z1)/2)=|mv|2d/2.subscript𝜑0𝑣superscriptsubscript121subscript2𝑡1superscript𝑑𝑡subscript𝑥1differential-d𝑥superscript𝑡𝑧12subscript𝑡1superscript𝑑𝑡subscript𝑥1differential-d𝑥superscript𝑡𝑧12subscript𝑚𝑣superscript2𝑑2\displaystyle\varphi_{0,v}(1_{2})^{-1}\bigg{(}\textstyle{\int}_{|2|\leqslant|t|\leqslant 1}d^{\times}t\textstyle{\int}_{|x|=1}dx|t|^{(z-1)/2}+\textstyle{\int}_{|t|=1}d^{\times}t\textstyle{\int}_{|x|<1}dx|t|^{(z-1)/2}\bigg{)}=|m_{v}|2^{-d/2}.

(iii) Suppose ordv(τ)=1subscriptord𝑣𝜏1\operatorname{ord}_{v}(\tau)=1. Then, |a||τ|𝑎𝜏|a|\leqslant|\tau| implies |a|<1𝑎1|a|<1, where |c2(a2τ)|=|c|2|τ|superscript𝑐2superscript𝑎2𝜏superscript𝑐2𝜏|c^{-2}(a^{2}-\tau)|=|c|^{-2}|\tau| is an odd power of q𝑞q and (10.14) is never attained. Thus 𝔈v(z)(γ^v)=0superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣0{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=0.

(II) The case |a|>|τ|𝑎𝜏|a|>|\tau|.

Since |τ|<|a2|𝜏superscript𝑎2|\tau|<|a^{2}|, the condition (10.14) yields |c|=|a|𝑐𝑎|c|=|a|. Thus the condition (10.12) is equivalent to a1x𝔬superscript𝑎1𝑥𝔬a^{-1}x\in\mathfrak{o}, a1t1(1τx2)𝔬superscript𝑎1superscript𝑡11𝜏superscript𝑥2𝔬a^{-1}t^{-1}(1-\tau x^{2})\in\mathfrak{o}, and the condition (10.13) to a1t𝔬superscript𝑎1𝑡𝔬a^{-1}t\in\mathfrak{o}. Hence,

(10.15) φ0,v(12)𝔈v(z)(γ^v)=|a|(z+1)/2tτ1𝔬{0}|t|(z1)/2vol(X(t))d×t.subscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣superscript𝑎𝑧12subscript𝑡superscript𝜏1𝔬0superscript𝑡𝑧12vol𝑋𝑡superscript𝑑𝑡\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=|a|^{(z+1)/2}\textstyle{\int}_{t\in\tau^{-1}{\mathfrak{o}}-\{0\}}|t|^{(z-1)/2}{\operatorname{vol}}(X(t))\,d^{\times}t.

where we put X(t)={xτ1𝔬|a2τx2t𝔬}𝑋𝑡conditional-set𝑥superscript𝜏1𝔬superscript𝑎2𝜏superscript𝑥2𝑡𝔬X(t)=\{x\in\tau^{-1}{\mathfrak{o}}\,|\,a^{-2}-\tau x^{2}\in t{\mathfrak{o}}\}. For l𝑙l\in{\mathbb{Z}}, we set

[ql/2]={ql/2,(l0(mod2)),q(l+1)/2,(l1(mod2)).delimited-[]superscript𝑞𝑙2casessuperscript𝑞𝑙2𝑙annotated0pmod2otherwisesuperscript𝑞𝑙12𝑙annotated1pmod2otherwise\displaystyle[q^{-l/2}]=\begin{cases}q^{-l/2},\quad(l\equiv 0\pmod{2}),\\ q^{-(l+1)/2},\quad(l\equiv 1\pmod{2}).\end{cases}
Lemma 10.2.

Let tτ1𝔬{0}𝑡superscript𝜏1𝔬0t\in\tau^{-1}{\mathfrak{o}}-\{0\}. If vΣdyadic𝑣subscriptΣdyadicv\not\in\Sigma_{\rm dyadic} and τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in{\mathfrak{o}}^{\times}-({\mathfrak{o}}^{\times})^{2}, we have

vol(X(t))=δ(|a2||t|)qd/2[|t|1/2].vol𝑋𝑡𝛿superscript𝑎2𝑡superscript𝑞𝑑2delimited-[]superscript𝑡12{\operatorname{vol}}(X(t))=\delta(|a^{-2}|\leqslant|t|)\,q^{-d/2}\,[|t|^{1/2}].

If vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} and τ=5𝜏5\tau=5,

vol(X(t))=δ(|4a2||t|)2d/2{[|t|1/2](|a2||t|),21|a1|(|4a2||t|<|a2|).vol𝑋𝑡𝛿4superscript𝑎2𝑡superscript2𝑑2casesdelimited-[]superscript𝑡12superscript𝑎2𝑡superscript21superscript𝑎14superscript𝑎2𝑡superscript𝑎2{\operatorname{vol}}(X(t))=\delta(|4a^{-2}|\leqslant|t|)2^{-d/2}\begin{cases}[|t|^{1/2}]&(|a^{-2}|\leqslant|t|),\\ 2^{-1}|a^{-1}|&(|4a^{-2}|\leqslant|t|<|a^{-2}|).\end{cases}

If τ𝔭𝔭2𝜏𝔭superscript𝔭2\tau\in{\mathfrak{p}}-{\mathfrak{p}}^{2}, we have vol(X(t))=δ(|a2||t|)[|ϖ1t|1/2]qvd/2vol𝑋𝑡𝛿superscript𝑎2𝑡delimited-[]superscriptsuperscriptitalic-ϖ1𝑡12superscriptsubscript𝑞𝑣𝑑2{\operatorname{vol}}(X(t))=\delta(|a^{-2}|\leqslant|t|)[|\varpi^{-1}t|^{1/2}]q_{v}^{-d/2}. If vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} and τ{5,1}𝜏51\tau\in\{-5,-1\}, we have

vol(X(t))=δ(|2a2||t|)2d/2{[|t|1/2](|a2||t|),21|a1|(|t|=|2a2|).vol𝑋𝑡𝛿2superscript𝑎2𝑡superscript2𝑑2casesdelimited-[]superscript𝑡12superscript𝑎2𝑡superscript21superscript𝑎1𝑡2superscript𝑎2{\operatorname{vol}}(X(t))=\delta(|2a^{-2}|\leqslant|t|)2^{-d/2}\begin{cases}[|t|^{1/2}]&(|a^{-2}|\leqslant|t|),\\ 2^{-1}|a^{-1}|&(|t|=|2a^{-2}|).\end{cases}
Proof.

In the proof, we write X𝑋X for X(t)𝑋𝑡X(t) for simplicity. First suppose τ𝔬×𝜏superscript𝔬\tau\in{\mathfrak{o}}^{\times}. Then xX𝑥𝑋x\in X is equivalent to x𝔬𝑥𝔬x\in\mathfrak{o} and a2τx2t𝔬superscript𝑎2𝜏superscript𝑥2𝑡𝔬a^{-2}-\tau x^{2}\in t{\mathfrak{o}}. Suppose vΣdyadic𝑣subscriptΣdyadicv\not\in\Sigma_{\rm dyadic}. Due to Lemma 7.1, Xa1𝔬={xa1𝔬|a2(1τ(ax)2)t𝔬}={xa1𝔬||a2||t|}𝑋superscript𝑎1𝔬conditional-set𝑥superscript𝑎1𝔬superscript𝑎21𝜏superscript𝑎𝑥2𝑡𝔬conditional-set𝑥superscript𝑎1𝔬superscript𝑎2𝑡X\cap a^{-1}{\mathfrak{o}}=\{x\in a^{-1}{\mathfrak{o}}\ |\ a^{-2}(1-\tau(ax)^{2})\in t{\mathfrak{o}}\}=\{x\in a^{-1}{\mathfrak{o}}\ |\ |a^{-2}|\leqslant|t|\} and Xa1𝔬={x𝔬||a1|<|x||t|1/2}𝑋superscript𝑎1𝔬conditional-set𝑥𝔬superscript𝑎1𝑥superscript𝑡12X-a^{-1}{\mathfrak{o}}=\{x\in{\mathfrak{o}}\ |\ |a^{-1}|<|x|\leqslant|t|^{1/2}\}; thus

vol(X)=vol𝑋absent\displaystyle{\operatorname{vol}}(X)= vol(Xa1𝔬)+vol(Xa1𝔬)vol𝑋superscript𝑎1𝔬vol𝑋superscript𝑎1𝔬\displaystyle{\operatorname{vol}}(X\cap a^{-1}{\mathfrak{o}})+{\operatorname{vol}}(X-a^{-1}{\mathfrak{o}})
=\displaystyle= δ(|a2||t|)|a1|qd/2+δ(|a1|2<|t|)([|t|1/2]|a1|)qd/2𝛿superscript𝑎2𝑡superscript𝑎1superscript𝑞𝑑2𝛿superscriptsuperscript𝑎12𝑡delimited-[]superscript𝑡12superscript𝑎1superscript𝑞𝑑2\displaystyle\delta(|a^{-2}|\leqslant|t|)|a^{-1}|q^{-d/2}+\delta(|a^{-1}|^{2}<|t|)([|t|^{1/2}]-|a^{-1}|)q^{-d/2}
=\displaystyle= δ(|a2||t|)qd/2[|t|1/2].𝛿superscript𝑎2𝑡superscript𝑞𝑑2delimited-[]superscript𝑡12\displaystyle\delta(|a^{-2}|\leqslant|t|)\,q^{-d/2}\,[|t|^{1/2}].

Suppose vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} and τ=5𝜏5\tau=5. Then since (2×)2=1+82superscriptsuperscriptsubscript2218subscript2({\mathbb{Z}}_{2}^{\times})^{2}=1+8{\mathbb{Z}}_{2}, we have 15(ax)242×15superscript𝑎𝑥24superscriptsubscript21-5(ax)^{2}\in 4{\mathbb{Z}}_{2}^{\times} for xX𝑥𝑋x\in X. Hence Xa1𝔬×𝑋superscript𝑎1superscript𝔬X\cap a^{-1}\mathfrak{o}^{\times} is empty unless 4a2t24superscript𝑎2𝑡subscript24a^{-2}\in t{\mathbb{Z}}_{2}, in which case Xa1𝔬×=a12×𝑋superscript𝑎1superscript𝔬superscript𝑎1superscriptsubscript2X\cap a^{-1}\mathfrak{o}^{\times}=a^{-1}\mathbb{Z}_{2}^{\times}. Similarly, Xa1ϖ𝔬𝑋superscript𝑎1italic-ϖ𝔬X\cap a^{-1}\varpi{\mathfrak{o}} is empty unless a2t2superscript𝑎2𝑡subscript2a^{-2}\in t{\mathbb{Z}}_{2}, in which case it is a122superscript𝑎12subscript2a^{-1}2{\mathbb{Z}}_{2}. We also have Xa1𝔬={x2||a1|<|x||t|1/2}.𝑋superscript𝑎1𝔬conditional-set𝑥subscript2superscript𝑎1𝑥superscript𝑡12X-a^{-1}{\mathfrak{o}}=\{x\in\mathbb{Z}_{2}\ |\ |a^{-1}|<|x|\leqslant|t|^{1/2}\}. Thus

vol(X)=vol𝑋absent\displaystyle{\operatorname{vol}}(X)= qd/2{δ(|4a2||t|)(1q1)|a1|+δ(|a2||t|)q1|a1|\displaystyle\,q^{-d/2}\{\delta(|4a^{-2}|\leqslant|t|)(1-q^{-1})|a^{-1}|+\delta(|a^{-2}|\leqslant|t|)q^{-1}|a^{-1}|
+δ(|a2||t|)([|t|1/2]|a1|)},\displaystyle+\delta(|a^{-2}|\leqslant|t|)([|t|^{1/2}]-|a^{-1}|)\},

which is simplified to the desired form. The case vΣdyadic𝑣subscriptΣdyadicv\in\Sigma_{\rm dyadic} with τ{5,1}𝜏51\tau\in\{-5,-1\} is similar.

Next suppose τ𝔭𝔭2𝜏𝔭superscript𝔭2\tau\in{\mathfrak{p}}-{\mathfrak{p}}^{2}. In the case, 1τu2𝔬×1𝜏superscript𝑢2superscript𝔬1-\tau u^{2}\in\mathfrak{o}^{\times} for all u𝔬v𝑢subscript𝔬𝑣u\in{\mathfrak{o}}_{v}. Hence Xa1𝔬={xa1𝔬|a2(1τa2x2)t𝔬}={xa1𝔬||a2||t|}𝑋superscript𝑎1𝔬conditional-set𝑥superscript𝑎1𝔬superscript𝑎21𝜏superscript𝑎2superscript𝑥2𝑡𝔬conditional-set𝑥superscript𝑎1𝔬superscript𝑎2𝑡X\cap a^{-1}{\mathfrak{o}}=\{x\in a^{-1}{\mathfrak{o}}\ |\ a^{-2}(1-\tau a^{2}x^{2})\in t{\mathfrak{o}}\}=\{x\in a^{-1}{\mathfrak{o}}\ |\ |a^{-2}|\leqslant|t|\} whose volume is δ(|a2||t|)|a1|qd/2𝛿superscript𝑎2𝑡superscript𝑎1superscript𝑞𝑑2\delta(|a^{-2}|\leqslant|t|)|a^{-1}|q^{-d/2}, and Xa1𝔬={xτ1𝔬||a1|<|x||τ1t|1/2}𝑋superscript𝑎1𝔬conditional-set𝑥superscript𝜏1𝔬superscript𝑎1𝑥superscriptsuperscript𝜏1𝑡12X-a^{-1}{\mathfrak{o}}=\{x\in\tau^{-1}{\mathfrak{o}}\ |\ |a^{-1}|<|x|\leqslant|\tau^{-1}t|^{1/2}\} whose volume is δ(|a|2|t|)([|ϖ1t|1/2]|a1|)qd/2𝛿superscript𝑎2𝑡delimited-[]superscriptsuperscriptitalic-ϖ1𝑡12superscript𝑎1superscript𝑞𝑑2\delta(|a|^{-2}\leqslant|t|)([|\varpi^{-1}t|^{1/2}]-|a^{-1}|)q^{-d/2}. By vol(X)=vol(Xa1𝔬)+vol(Xa1𝔬)vol𝑋vol𝑋superscript𝑎1𝔬vol𝑋superscript𝑎1𝔬{\operatorname{vol}}(X)={\operatorname{vol}}(X\cap a^{-1}{\mathfrak{o}})+{\operatorname{vol}}(X-a^{-1}\mathfrak{o}), we are done. ∎

Lemma 10.3.

For any z𝑧z\in\mathbb{C} such that qzq±1superscript𝑞𝑧superscript𝑞plus-or-minus1q^{z}\neq q^{\pm 1} and aFv𝑎subscript𝐹𝑣a\in F_{v} such that |a|>|τ|𝑎𝜏|a|>|\tau|, we have

|a2||t||τ1|[|τ1t|1/2]|t|(z1)/2d×t=qd/2{1+q(1)δz+121qz+1+qz(1)δ21qz|a|z}subscriptsuperscript𝑎2𝑡superscript𝜏1delimited-[]superscriptsuperscript𝜏1𝑡12superscript𝑡𝑧12superscript𝑑𝑡superscript𝑞𝑑21superscript𝑞superscript1𝛿𝑧121superscript𝑞𝑧1superscript𝑞𝑧superscript1𝛿21superscript𝑞𝑧superscript𝑎𝑧\displaystyle\int_{|a^{-2}|\leqslant|t|\leqslant|\tau^{-1}|}[|\tau^{-1}t|^{1/2}]\,|t|^{(z-1)/2}d^{\times}t=q^{-d/2}\left\{\frac{1+q^{-(-1)^{\delta}\frac{z+1}{2}}}{1-q^{-z}}+\frac{1+q^{\frac{z-(-1)^{\delta}}{2}}}{1-q^{z}}|a|^{-z}\right\}

with δ=ordv(τ){0,1}𝛿subscriptord𝑣𝜏01\delta=\operatorname{ord}_{v}(\tau)\in\{0,1\}.

Proof.

A direct computation. ∎

First we consider the case where τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in{\mathfrak{o}}^{\times}-({\mathfrak{o}}^{\times})^{2}. If v𝑣v is non-dyadic, from (10.15), Lemmas 10.2 and 10.3, we get

φv,0(12)𝔈v(z)(γ^v)=subscript𝜑𝑣0subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣absent\displaystyle\varphi_{v,0}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})= =qd|a|(z+1)/2(1+q(z1)/21qz+1+q(z1)/21qz|a|z).absentsuperscript𝑞𝑑superscript𝑎𝑧121superscript𝑞𝑧121superscript𝑞𝑧1superscript𝑞𝑧121superscript𝑞𝑧superscript𝑎𝑧\displaystyle=q^{-d}|a|^{(z+1)/2}\left(\tfrac{1+q^{(-z-1)/2}}{1-q^{-z}}+\tfrac{1+q^{(z-1)/2}}{1-q^{z}}|a|^{-z}\right).

To complete the proof, it suffices to use Lemma 7.4 and to note that |nmv2|𝑛superscriptsubscript𝑚𝑣2|\tfrac{n}{m_{v}^{2}}| equals |a|2superscript𝑎2|a|^{2} or 111 according to |a|>1𝑎1|a|>1 or |a|1𝑎1|a|\leqslant 1, which follows from Lemma 7.11 (1) and Proposition 7.12 (1). If v𝑣v is dyadic and τ=5𝜏5\tau=5, in the same way, we have

φ0,v(12)𝔈v(z)(γ^v)subscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) =|a|(z+1)/22d/2{|4a2||t|<|a2|21|a|1|t|(z1)/2d×t+|a|2|t|1[|t|1/2]|t|(z1)/2d×t}absentsuperscript𝑎𝑧12superscript2𝑑2subscript4superscript𝑎2𝑡superscript𝑎2superscript21superscript𝑎1superscript𝑡𝑧12superscript𝑑𝑡subscriptsuperscript𝑎2𝑡1delimited-[]superscript𝑡12superscript𝑡𝑧12superscript𝑑𝑡\displaystyle=|a|^{(z+1)/2}2^{-d/2}\left\{\textstyle{\int}_{|4a^{-2}|\leqslant|t|<|a^{-2}|}2^{-1}|a|^{-1}|t|^{(z-1)/2}{{d}}^{\times}t+\int_{|a|^{-2}\leqslant|t|\leqslant 1}[|t|^{1/2}]|t|^{(z-1)/2}{{d}}^{\times}t\right\}
=2d(1+2(z1)/212z|a|z+12+1+2(z1)/22z(12z)|a|(z+1)/2).absentsuperscript2𝑑1superscript2𝑧121superscript2𝑧superscript𝑎𝑧121superscript2𝑧12superscript2𝑧1superscript2𝑧superscript𝑎𝑧12\displaystyle=2^{-d}\left(\tfrac{1+2^{(-z-1)/2}}{1-2^{-z}}|a|^{\frac{z+1}{2}}+\tfrac{1+2^{(z-1)/2}}{2^{z}(1-2^{z})}|a|^{(-z+1)/2}\right).

To complete the proof, it suffices to use Lemma 7.4 and to note that |nmv2|=|a|2𝑛superscriptsubscript𝑚𝑣2superscript𝑎2|\tfrac{n}{m_{v}^{2}}|=|a|^{-2} if |a|>1𝑎1|a|>1 from Lemma 7.11 (4) and Proposition 7.12 (1). The remaining cases are similar.

10.2.3. The proof of Theorem 7.9 (4)

Let vS(𝔫)𝑣𝑆𝔫v\in S({\mathfrak{n}}) and Φv=chZv𝕂0(𝔭)subscriptΦ𝑣subscriptchsubscript𝑍𝑣subscript𝕂0𝔭\Phi_{v}={\rm ch}_{Z_{v}{\mathbb{K}}_{0}({\mathfrak{p}})}. By (3.10) and (10.11), we have

(10.16) φ0,v(12)𝔈v(z)(γ^v)=vol(𝐊0(𝔭))|a|(z+1)/2Fv×{B(t)+ξ𝔬v/𝔭vBξ(t)}|t|(z1)/2d×t,subscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣volsubscript𝐊0𝔭superscript𝑎𝑧12subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡subscript𝜉subscript𝔬𝑣subscript𝔭𝑣subscript𝐵𝜉𝑡superscript𝑡𝑧12superscript𝑑𝑡\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})={\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}))|a|^{(z+1)/2}\int_{F_{v}^{\times}}\{B(t)+\sum_{\xi\in{\mathfrak{o}}_{v}/{\mathfrak{p}}_{v}}B_{\xi}(t)\}|t|^{(z-1)/2}d^{\times}t,

with

B(t)𝐵𝑡\displaystyle B(t) =FvΦv([1τxt1(a2τx2)τt1+τx])𝑑x,Bξ(t)=FvΦv([1+τ(x+ξt)τtt1(a2τ(x+ξt)2)1τ(x+tξ)])𝑑x.formulae-sequenceabsentsubscriptsubscript𝐹𝑣subscriptΦ𝑣delimited-[]1𝜏𝑥superscript𝑡1superscript𝑎2𝜏superscript𝑥2𝜏𝑡1𝜏𝑥differential-d𝑥subscript𝐵𝜉𝑡subscriptsubscript𝐹𝑣subscriptΦ𝑣delimited-[]1𝜏𝑥𝜉𝑡𝜏𝑡superscript𝑡1superscript𝑎2𝜏superscript𝑥𝜉𝑡21𝜏𝑥𝑡𝜉differential-d𝑥\displaystyle=\int_{F_{v}}\Phi_{v}(\left[\begin{smallmatrix}1-\tau x&t^{-1}(a^{-2}-\tau x^{2})\\ \tau t&1+\tau x\end{smallmatrix}\right])dx,\quad B_{\xi}(t)=\int_{F_{v}}\Phi_{v}([\begin{smallmatrix}1+\tau(x+\xi t)&-\tau t\\ -t^{-1}(a^{-2}-\tau(x+\xi t)^{2})&1-\tau(x+t\xi)\end{smallmatrix}])dx.

We consider the case |a||τ|𝑎𝜏|a|\leqslant|\tau|.

Lemma 10.4.

Suppose |a||τ|𝑎𝜏|a|\leqslant|\tau|. Then, we have B(t)=Bξ(t)=0𝐵𝑡subscript𝐵𝜉𝑡0B(t)=B_{\xi}(t)=0.

Proof.

Suppose τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in{\mathfrak{o}}^{\times}-({\mathfrak{o}}^{\times})^{2}. Then |a|1𝑎1|a|\leqslant 1. We have that Bξ(t)0subscript𝐵𝜉𝑡0B_{\xi}(t)\not=0 if and only if there exists cFv×𝑐superscriptsubscript𝐹𝑣c\in F_{v}^{\times} with the properties:

(10.17) c(1±τy)𝔬,cτt𝔬,cτt1(a2τy2)𝔭,c2(1τa2)𝔬×,formulae-sequence𝑐plus-or-minus1𝜏𝑦𝔬formulae-sequence𝑐𝜏𝑡𝔬formulae-sequence𝑐𝜏superscript𝑡1superscript𝑎2𝜏superscript𝑦2𝔭superscript𝑐21𝜏superscript𝑎2superscript𝔬\displaystyle c(1\pm\tau y)\in\mathfrak{o},\quad c\tau t\in\mathfrak{o},\quad c\tau t^{-1}(a^{-2}-\tau y^{2})\in{\mathfrak{p}},\quad c^{2}(1-\tau a^{-2})\in\mathfrak{o}^{\times},

where we set y=x+ξt𝑦𝑥𝜉𝑡y=x+\xi t. By |a|1𝑎1|a|\leqslant 1, then |1τa2|=|a|21𝜏superscript𝑎2superscript𝑎2|1-\tau a^{-2}|=|a|^{-2}, and hence the last condition of (10.17) yields |c|=|a|𝑐𝑎|c|=|a|. From the first and the second ones, we then have ay𝔬𝑎𝑦𝔬ay\in\mathfrak{o} and at𝔬𝑎𝑡𝔬at\in\mathfrak{o}. The third condition is impossible by Lemma 7.1. Thus Bξ(t)=0subscript𝐵𝜉𝑡0B_{\xi}(t)=0 for all tFv𝑡subscript𝐹𝑣t\in F_{v}. In the same way, B(t)=0𝐵𝑡0B(t)=0 for all tFv𝑡subscript𝐹𝑣t\in F_{v}. Suppose ordv(τ)=1subscriptord𝑣𝜏1\operatorname{ord}_{v}(\tau)=1. Then, |a||ϖ|=q1𝑎italic-ϖsuperscript𝑞1|a|\leqslant|\varpi|=q^{-1}, and B~ξ(t)0subscript~𝐵𝜉𝑡0\tilde{B}_{\xi}(t)\not=0 if and only if (10.17); from the last of (10.17), 1=|c2(1τa2)|=|c|2|τ||a|21superscript𝑐21𝜏superscript𝑎2superscript𝑐2𝜏superscript𝑎21=|c^{2}(1-\tau a^{-2})|=|c|^{2}|\tau||a|^{-2}, which is impossible due to |τ|=q1𝜏superscript𝑞1|\tau|=q^{-1}. Hence B~ξ(t)=0subscript~𝐵𝜉𝑡0{\tilde{B}}_{\xi}(t)=0 for all tFv×𝑡superscriptsubscript𝐹𝑣t\in F_{v}^{\times}. Similarly, we obtain B(t)=0𝐵𝑡0B(t)=0 for all tFv×𝑡superscriptsubscript𝐹𝑣t\in F_{v}^{\times}. ∎

Next we consider the case |a|>|τ|𝑎𝜏|a|>|\tau|. For the computation of B(t)𝐵𝑡B(t) and Bξ(t)subscript𝐵𝜉𝑡B_{\xi}(t), we set X(t)={xτ1𝔬|a2τx2t𝔬}𝑋𝑡conditional-set𝑥superscript𝜏1𝔬superscript𝑎2𝜏superscript𝑥2𝑡𝔬X(t)=\{x\in\tau^{-1}{\mathfrak{o}}|a^{-2}-\tau x^{2}\in t{\mathfrak{o}}\}.

Lemma 10.5.

If |a|>|τ|𝑎𝜏|a|>|\tau|,

B(t)=δ(tτ1𝔭)vol(X(t)),Bξ(ϖ1t)=δ(ϖ1tτ1𝔬)vol(X(t)).formulae-sequence𝐵𝑡𝛿𝑡superscript𝜏1𝔭vol𝑋𝑡subscript𝐵𝜉superscriptitalic-ϖ1𝑡𝛿superscriptitalic-ϖ1𝑡superscript𝜏1𝔬vol𝑋𝑡\displaystyle B(t)=\delta(t\in\tau^{-1}{\mathfrak{p}}){\operatorname{vol}}(X(t)),\quad B_{\xi}(\varpi^{-1}t)=\delta(\varpi^{-1}t\in\tau^{-1}{\mathfrak{o}}){\operatorname{vol}}(X(t)).
Proof.

The integrand of B(t)𝐵𝑡B(t) is non-zero if and only if there exists cFv×𝑐superscriptsubscript𝐹𝑣c\in F_{v}^{\times} such that 1τxc𝔬1𝜏𝑥𝑐𝔬1-\tau x\in c{\mathfrak{o}}, a2τx2ct𝔬superscript𝑎2𝜏superscript𝑥2𝑐𝑡𝔬a^{-2}-\tau x^{2}\in ct{\mathfrak{o}}, τtc𝔭𝜏𝑡𝑐𝔭\tau t\in c{\mathfrak{p}}, 1+τxc𝔬1𝜏𝑥𝑐𝔬1+\tau x\in c{\mathfrak{o}}, 1τa2c2𝔬×1𝜏superscript𝑎2superscript𝑐2superscript𝔬1-\tau a^{-2}\in c^{2}{\mathfrak{o}}^{\times}. By |a|>|τ|𝑎𝜏|a|>|\tau|, 1τa21𝜏superscript𝑎21-\tau a^{-2} is a unit and whence |c|=1𝑐1|c|=1. From this, B(t)=δ(tτ1𝔭)xτ1𝔬,a2τx2t𝔬𝑑x𝐵𝑡𝛿𝑡superscript𝜏1𝔭subscriptformulae-sequence𝑥superscript𝜏1𝔬superscript𝑎2𝜏superscript𝑥2𝑡𝔬differential-d𝑥B(t)=\delta(t\in\tau^{-1}{\mathfrak{p}})\int_{x\in\tau^{-1}{\mathfrak{o}},a^{-2}-\tau x^{2}\in t{\mathfrak{o}}}dx.

The integrand of Bξ(t)subscript𝐵𝜉𝑡B_{\xi}(t) is non-zero if and only if 1+τ(x+ξt)c𝔬1𝜏𝑥𝜉𝑡𝑐𝔬1+\tau(x+\xi t)\in c{\mathfrak{o}}, τt𝔬𝜏𝑡𝔬\tau t\in{\mathfrak{o}}, a2τ(x+ξt)2ct𝔭superscript𝑎2𝜏superscript𝑥𝜉𝑡2𝑐𝑡𝔭a^{-2}-\tau(x+\xi t)^{2}\in ct{\mathfrak{p}}, 1τ(x+ξt)c𝔬1𝜏𝑥𝜉𝑡𝑐𝔬1-\tau(x+\xi t)\in c{\mathfrak{o}}, 1τa2c2𝔬×1𝜏superscript𝑎2superscript𝑐2superscript𝔬1-\tau a^{-2}\in c^{2}{\mathfrak{o}}^{\times}. By |a|>1𝑎1|a|>1, c𝑐c is a unit and whence

Bξ(t)=subscript𝐵𝜉𝑡absent\displaystyle B_{\xi}(t)= δ(tτ1𝔬)xτ1𝔬,a2τ(x+ξt)2ϖt𝔬𝑑x=δ(tτ1𝔬)xτ1𝔬,a2τx2ϖt𝔬𝑑x𝛿𝑡superscript𝜏1𝔬subscriptformulae-sequence𝑥superscript𝜏1𝔬superscript𝑎2𝜏superscript𝑥𝜉𝑡2italic-ϖ𝑡𝔬differential-d𝑥𝛿𝑡superscript𝜏1𝔬subscriptformulae-sequence𝑥superscript𝜏1𝔬superscript𝑎2𝜏superscript𝑥2italic-ϖ𝑡𝔬differential-d𝑥\displaystyle\delta(t\in\tau^{-1}{\mathfrak{o}})\textstyle{\int}_{x\in\tau^{-1}{\mathfrak{o}},\,a^{-2}-\tau(x+\xi t)^{2}\in\varpi t{\mathfrak{o}}}dx=\delta(t\in\tau^{-1}{\mathfrak{o}})\textstyle{\int}_{x\in\tau^{-1}{\mathfrak{o}},a^{-2}-\tau x^{2}\in\varpi t{\mathfrak{o}}}dx

Thus, if τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in{\mathfrak{o}}^{\times}-({\mathfrak{o}}^{\times})^{2}, the value (10.16) for |a|1𝑎1|a|\leqslant 1 vanishes by Lemma 10.4. For |a|>1𝑎1|a|>1, Lemmas 10.2, 10.3 and 10.5 yield an evaluation of (10.16) as

vol(𝐊0(𝔭))|a|(z+1)/2(1+q(z+1)/2)τ1𝔭{0}vol(X(t))|t|(z1)/2d×tvolsubscript𝐊0𝔭superscript𝑎𝑧121superscript𝑞𝑧12subscriptsuperscript𝜏1𝔭0vol𝑋𝑡superscript𝑡𝑧12superscript𝑑𝑡\displaystyle{\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}))|a|^{(z+1)/2}(1+q^{(z+1)/2})\textstyle{\int}_{\tau^{-1}{\mathfrak{p}}-\{0\}}{\operatorname{vol}}(X(t))|t|^{(z-1)/2}d^{\times}t
=\displaystyle= vol(𝐊0(𝔭))|a|(z+1)/2(1+q(z+1)/2)(𝔬{0}vol(X(t))|t|(z1)/2d×t𝔬×qd/2|t|(z1)/2d×t)volsubscript𝐊0𝔭superscript𝑎𝑧121superscript𝑞𝑧12subscript𝔬0vol𝑋𝑡superscript𝑡𝑧12superscript𝑑𝑡subscriptsuperscript𝔬superscript𝑞𝑑2superscript𝑡𝑧12superscript𝑑𝑡\displaystyle{\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}))|a|^{(z+1)/2}(1+q^{(z+1)/2})\left(\textstyle{\int}_{{\mathfrak{o}}-\{0\}}{\operatorname{vol}}(X(t))|t|^{(z-1)/2}d^{\times}t-\textstyle{\int}_{{\mathfrak{o}}^{\times}}q^{-d/2}|t|^{(z-1)/2}d^{\times}t\right)
=\displaystyle= |a|(z+1)/21+q(z+1)/21+q(𝔬{0}vol(X(t))|t|(z1)/2d×tqd)superscript𝑎𝑧121superscript𝑞𝑧121𝑞subscript𝔬0vol𝑋𝑡superscript𝑡𝑧12superscript𝑑𝑡superscript𝑞𝑑\displaystyle|a|^{(z+1)/2}\frac{1+q^{(z+1)/2}}{1+q}\left(\textstyle{\int}_{{\mathfrak{o}}-\{0\}}{\operatorname{vol}}(X(t))|t|^{(z-1)/2}d^{\times}t-q^{-d}\right)
=\displaystyle= qd(1+q(z1)/21qz1+q(z+1)/21+q|a|(z+1)/2+1+q(z1)/21qz1+q(z+1)/21+q|a|(z+1)/2).superscript𝑞𝑑1superscript𝑞𝑧121superscript𝑞𝑧1superscript𝑞𝑧121𝑞superscript𝑎𝑧121superscript𝑞𝑧121superscript𝑞𝑧1superscript𝑞𝑧121𝑞superscript𝑎𝑧12\displaystyle q^{-d}\left(\frac{1+q^{(-z-1)/2}}{1-q^{-z}}\frac{1+q^{(-z+1)/2}}{1+q}|a|^{(z+1)/2}+\frac{1+q^{(z-1)/2}}{1-q^{z}}\frac{1+q^{(z+1)/2}}{1+q}|a|^{(-z+1)/2}\right).

If ordv(τ)=1subscriptord𝑣𝜏1\operatorname{ord}_{v}(\tau)=1, the value (10.16) for |a|1𝑎1|a|\geqslant 1 is evaluated as

vol(𝐊0(𝔭))|a|(z+1)/2(1+q(z+1)/2){τ1𝔬{0}vol(X(t))|t|(z1)/2d×tτ1𝔬×qd/2q|t|(z1)/2d×t}volsubscript𝐊0𝔭superscript𝑎𝑧121superscript𝑞𝑧12subscriptsuperscript𝜏1𝔬0vol𝑋𝑡superscript𝑡𝑧12superscript𝑑𝑡subscriptsuperscript𝜏1superscript𝔬superscript𝑞𝑑2𝑞superscript𝑡𝑧12superscript𝑑𝑡\displaystyle{\operatorname{vol}}({\mathbf{K}}_{0}({\mathfrak{p}}))|a|^{(z+1)/2}(1+q^{(z+1)/2})\{\textstyle{\int}_{\tau^{-1}{\mathfrak{o}}-\{0\}}{\operatorname{vol}}(X(t))|t|^{(z-1)/2}d^{\times}t-\textstyle{\int}_{\tau^{-1}{\mathfrak{o}}^{\times}}q^{-d/2}q|t|^{(z-1)/2}d^{\times}t\}
=\displaystyle= (1+q(z+1)/2)×qd(11qz1+q(z+1)/21+q|a|(z+1)/2+11qz1+q(z+1)/21+q|a|(z+1)/2).1superscript𝑞𝑧12superscript𝑞𝑑11superscript𝑞𝑧1superscript𝑞𝑧121𝑞superscript𝑎𝑧1211superscript𝑞𝑧1superscript𝑞𝑧121𝑞superscript𝑎𝑧12\displaystyle(1+q^{(z+1)/2})\times q^{-d}\left(\frac{1}{1-q^{-z}}\frac{1+q^{(-z+1)/2}}{1+q}|a|^{(z+1)/2}+\frac{1}{1-q^{z}}\frac{1+q^{(z+1)/2}}{1+q}|a|^{(-z+1)/2}\right).

Theorem 7.9 (4) follows from these as before by Lemmas 7.4, Lemma 7.11 and Proposition 7.12.

10.2.4. The proof of Theorem 7.9 (5)

Let vS𝑣𝑆v\in S and Φv(gv)=Φ(s;gv)subscriptΦ𝑣subscript𝑔𝑣Φ𝑠subscript𝑔𝑣\Phi_{v}(g_{v})=\Phi(s;g_{v}) the Green function on Gvsubscript𝐺𝑣G_{v} defined in §2.3. First we consider the case a0𝑎0a\neq 0. The case a=0𝑎0a=0 is treated after Lemma 10.12. From (2.3), the integral (10.11) is written as

φ0,v(12)𝔈v(z)(γ^v)=subscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣absent\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})= 𝔗Δ,v\GvΦ(s;g1γ^vg)φ0,v(g)𝑑gsubscript\subscript𝔗Δ𝑣subscript𝐺𝑣Φ𝑠superscript𝑔1subscript^𝛾𝑣𝑔subscript𝜑0𝑣𝑔differential-d𝑔\displaystyle\int_{{\mathfrak{T}}_{\Delta,v}\backslash G_{v}}\Phi(s;g^{-1}\hat{\gamma}_{v}g)\varphi_{0,v}(g)dg
=\displaystyle= (qs+12qs+12)1Fv×Fv|1b2τ|s+12superscriptsuperscript𝑞𝑠12superscript𝑞𝑠121subscriptsuperscriptsubscript𝐹𝑣subscriptsubscript𝐹𝑣superscript1superscript𝑏2𝜏𝑠12\displaystyle(q^{-\frac{s+1}{2}}-q^{\frac{s+1}{2}})^{-1}\,\textstyle{\int}_{F_{v}^{\times}}\textstyle{\int}_{F_{v}}|1-b^{2}\tau|^{\frac{s+1}{2}}
×{max(|1τbx|,|1+τbx|,|τbt|,|t1b(1τx2)|)2}s+12|t|z12d×tdx,\displaystyle\times\{\max(|1-\tau bx|,|1+\tau bx|,|\tau bt|,|t^{-1}b(1-\tau x^{2})|)^{2}\}^{-\frac{s+1}{2}}|t|^{\frac{z-1}{2}}{{d}}^{\times}t\,{{d}}x,

where b=a1𝑏superscript𝑎1b=a^{-1}. By the variable change tt/(τb)𝑡𝑡𝜏𝑏t\rightarrow t/(\tau b), xx/(τb)𝑥𝑥𝜏𝑏x\rightarrow x/(\tau b), we have

(10.18) φ0,v(12)𝔈v(z)(γ^v)=|1b2τ|s+12|τb|z+12(qs+12qs+12)1Fv×B(t)|t|z12d×tsubscript𝜑0𝑣subscript12superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣superscript1superscript𝑏2𝜏𝑠12superscript𝜏𝑏𝑧12superscriptsuperscript𝑞𝑠12superscript𝑞𝑠121subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡superscript𝑡𝑧12superscript𝑑𝑡\displaystyle\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v})=|1-b^{2}\tau|^{\frac{s+1}{2}}|\tau b|^{-\frac{z+1}{2}}(q^{-\frac{s+1}{2}}-q^{\frac{s+1}{2}})^{-1}\textstyle{\int}_{F_{v}^{\times}}B(t)|t|^{\frac{z-1}{2}}\,{{d}}^{\times}t

with

B(t)=Fvmax(|1x|,|1+x|,|t|,|t1(b2τx2)|)s1dx.B(t)=\textstyle{\int}_{F_{v}}\max(|1-x|,|1+x|,|t|,|t^{-1}(b^{2}\tau-x^{2})|)^{-s-1}\,{{d}}x.

When t𝑡t is viewed as a constant, we let f(x)𝑓𝑥f(x) denote the integrand of B(t)𝐵𝑡B(t). First we consider the case when τ=Δv0𝜏superscriptsubscriptΔ𝑣0\tau=\Delta_{v}^{0} is a non-unit.

Lemma 10.6.

Suppose |τ|=q1𝜏superscript𝑞1|\tau|=q^{-1}, |a|1𝑎1|a|\leqslant 1 and set b=a1𝑏superscript𝑎1b=a^{-1}. Let Re(s)>1/2Re𝑠12\operatorname{Re}(s)>-1/2.

  • (i)

    When |τb||t|𝜏𝑏𝑡|\tau b|\geqslant|t|,

    B(t)𝐵𝑡\displaystyle B(t) =qd/2|t|s+1|τb|2s1{qs1+(1q1)q2s1(1q2s1)1}.absentsuperscript𝑞𝑑2superscript𝑡𝑠1superscript𝜏𝑏2𝑠1superscript𝑞𝑠11superscript𝑞1superscript𝑞2𝑠1superscript1superscript𝑞2𝑠11\displaystyle=q^{-d/2}|t|^{s+1}|\tau b|^{-2s-1}\left\{q^{-s-1}+(1-q^{-1}){q^{-2s-1}}{(1-q^{-2s-1})^{-1}}\right\}.
  • (ii)

    When |τb|<|t|𝜏𝑏𝑡|\tau b|<|t|,

    B(t)𝐵𝑡\displaystyle B(t) =qd/2|t|s{1+(1q1)q2s1(1q2s1)1}.absentsuperscript𝑞𝑑2superscript𝑡𝑠11superscript𝑞1superscript𝑞2𝑠1superscript1superscript𝑞2𝑠11\displaystyle=q^{-d/2}|t|^{-s}\left\{1+(1-q^{-1}){q^{-2s-1}}{(1-q^{-2s-1})^{-1}}\right\}.
Proof.

Suppose |a|1𝑎1|a|\leqslant 1. Then |b|=|a|11𝑏superscript𝑎11|b|=|a|^{-1}\geqslant 1 and |τ|=q1𝜏superscript𝑞1|\tau|=q^{-1}. Hence |τb|q1𝜏𝑏superscript𝑞1|\tau b|\geqslant q^{-1}. By dividing the integration domain into D1={xFv||x|max(|τb|,|t|)}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝜏𝑏𝑡D_{1}=\{x\in F_{v}||x|\leqslant\max(|\tau b|,|t|)\} and into D2={xFv||x|>max(|τb|,|t|)}subscript𝐷2conditional-set𝑥subscript𝐹𝑣𝑥𝜏𝑏𝑡D_{2}=\{x\in F_{v}||x|>\max(|\tau b|,|t|)\}, we write the integral as the sum B1(t)+B2(t)subscript𝐵1𝑡subscript𝐵2𝑡B_{1}(t)+B_{2}(t) with Bi(t)=Dif(x)𝑑xsubscript𝐵𝑖𝑡subscriptsubscript𝐷𝑖𝑓𝑥differential-d𝑥B_{i}(t)=\int_{D_{i}}f(x){{d}}x. Let xD2𝑥subscript𝐷2x\in D_{2}. Then by |τb|q1𝜏𝑏superscript𝑞1|\tau b|\geqslant q^{-1}, we have |x|>max(|τb|,|t|)max(q1,|t|)q1𝑥𝜏𝑏𝑡superscript𝑞1𝑡superscript𝑞1|x|>\max(|\tau b|,|t|)\geqslant\max(q^{-1},|t|)\geqslant q^{-1}. Hence |1±x||x|plus-or-minus1𝑥𝑥|1\pm x|\leqslant|x|, and |x|2>|τb|2superscript𝑥2superscript𝜏𝑏2|x|^{2}>|\tau b|^{2} which implies |x|2|τb|2q=|τb2|superscript𝑥2superscript𝜏𝑏2𝑞𝜏superscript𝑏2|x|^{2}\geqslant|\tau b|^{2}q=|\tau b^{2}|. Hence |τb2x2|=|x|2𝜏superscript𝑏2superscript𝑥2superscript𝑥2|\tau b^{2}-x^{2}|=|x|^{2}. Thus f(x)=(|t|1|x|2)s1𝑓𝑥superscriptsuperscript𝑡1superscript𝑥2𝑠1f(x)=(|t|^{-1}|x|^{2})^{-s-1} for all xD2𝑥subscript𝐷2x\in D_{2} and

(10.19) B2(t)=xD2(|t|1|x|2)s1𝑑x=|t|s+1D2|x|2s2𝑑x.subscript𝐵2𝑡subscript𝑥subscript𝐷2superscriptsuperscript𝑡1superscript𝑥2𝑠1differential-d𝑥superscript𝑡𝑠1subscriptsubscript𝐷2superscript𝑥2𝑠2differential-d𝑥\displaystyle B_{2}(t)=\textstyle{\int}_{x\in D_{2}}(|t|^{-1}|x|^{2})^{-s-1}\,{{d}}x=|t|^{s+1}\textstyle{\int}_{D_{2}}|x|^{-2s-2}{{d}}x.

(i) Suppose |t||τb|𝑡𝜏𝑏|t|\leqslant|\tau b|. Then D1={xFv||x||τb|}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝜏𝑏D_{1}=\{x\in F_{v}||x|\leqslant|\tau b|\} and |t||τb||τb2|𝑡𝜏𝑏𝜏superscript𝑏2|t|\leqslant|\tau b|\leqslant|\tau b^{2}|. For x𝔬D1𝑥𝔬subscript𝐷1x\in\mathfrak{o}\cap D_{1}, we have |1±x|1|t|1|τb||t|1|τb2|=|t1(τb2x2)|plus-or-minus1𝑥1superscript𝑡1𝜏𝑏superscript𝑡1𝜏superscript𝑏2superscript𝑡1𝜏superscript𝑏2superscript𝑥2|1\pm x|\leqslant 1\leqslant|t|^{-1}|\tau b|\leqslant|t|^{-1}|\tau b^{2}|=|t^{-1}(\tau b^{2}-x^{2})| and |t||t|1|τ2b2|<|t|1|τb2|=|t1(b2τx2)|𝑡superscript𝑡1superscript𝜏2superscript𝑏2superscript𝑡1𝜏superscript𝑏2superscript𝑡1superscript𝑏2𝜏superscript𝑥2|t|\leqslant|t|^{-1}|\tau^{2}b^{2}|<|t|^{-1}|\tau b^{2}|=|t^{-1}(b^{2}\tau-x^{2})|. Hence f(x)=(|t|1|τb2|)s1𝑓𝑥superscriptsuperscript𝑡1𝜏superscript𝑏2𝑠1f(x)=(|t|^{-1}|\tau b^{2}|)^{-s-1}. For x(F𝔬)D1𝑥𝐹𝔬subscript𝐷1x\in(F-\mathfrak{o})\cap D_{1}, we have |1±x|=|x|plus-or-minus1𝑥𝑥|1\pm x|=|x| and |x||τb|𝑥𝜏𝑏|x|\leqslant|\tau b|. Hence |xt||τ2b2|<|τb2|𝑥𝑡superscript𝜏2superscript𝑏2𝜏superscript𝑏2|xt|\leqslant|\tau^{2}b^{2}|<|\tau b^{2}|. Thus f(x)=max(|x|,|t|,|t|1|τb2|)s1=(|t|1|τb2|)s1f(x)=\max(|x|,|t|,|t|^{-1}|\tau b^{2}|)^{-s-1}=(|t|^{-1}|\tau b^{2}|)^{-s-1}. Therefore,

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =|x||τb|(|t|1|τb2|)s1𝑑x=qd/2|τb|(|t|1|τb2|)s1.absentsubscript𝑥𝜏𝑏superscriptsuperscript𝑡1𝜏superscript𝑏2𝑠1differential-d𝑥superscript𝑞𝑑2𝜏𝑏superscriptsuperscript𝑡1𝜏superscript𝑏2𝑠1\displaystyle=\textstyle{\int}_{|x|\leqslant|\tau b|}(|t|^{-1}|\tau b^{2}|)^{-s-1}{{d}}x=q^{-d/2}|\tau b|(|t|^{-1}|\tau b^{2}|)^{-s-1}.

We have

B2(t)subscript𝐵2𝑡\displaystyle B_{2}(t) =|t|s+1|x|>|τb||x|2s2𝑑x=qd/2(1q1)|τb|2s1|t|s+1q2s11q2s1.absentsuperscript𝑡𝑠1subscript𝑥𝜏𝑏superscript𝑥2𝑠2differential-d𝑥superscript𝑞𝑑21superscript𝑞1superscript𝜏𝑏2𝑠1superscript𝑡𝑠1superscript𝑞2𝑠11superscript𝑞2𝑠1\displaystyle=|t|^{s+1}\textstyle{\int}_{|x|>|\tau b|}|x|^{-2s-2}{{d}}x=q^{-d/2}(1-q^{-1})|\tau b|^{-2s-1}|t|^{s+1}\frac{q^{-2s-1}}{1-q^{-2s-1}}.

(ii) Suppose |t|>|τb|𝑡𝜏𝑏|t|>|\tau b|. Then from |τ|=q1𝜏superscript𝑞1|\tau|=q^{-1}, we have |t||b|(1)𝑡annotated𝑏absent1|t|\geqslant|b|\,(\geqslant 1). Thus |t|2|b|2>|τb2|superscript𝑡2superscript𝑏2𝜏superscript𝑏2|t|^{2}\geqslant|b|^{2}>|\tau b^{2}|. Hence |t|>|t|1|τb2|𝑡superscript𝑡1𝜏superscript𝑏2|t|>|t|^{-1}|\tau b^{2}|. Moreover, D1={xFv||x||t|}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝑡D_{1}=\{x\in F_{v}||x|\leqslant|t|\,\}. If xD1𝑥subscript𝐷1x\in D_{1}, we have |1±x|max(1,|x|)|t|plus-or-minus1𝑥1𝑥𝑡|1\pm x|\leqslant\max(1,|x|)\leqslant|t| and |t1(τb2x2)|=max(|τb2|,|x2|)|t1|max(|t|2,|x|2)=|t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑡2superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|=\max(|\tau b^{2}|,|x^{2}|)\leqslant|t^{-1}|\max(|t|^{2},|x|^{2})=|t|; thus f(x)=|t|s1𝑓𝑥superscript𝑡𝑠1f(x)=|t|^{-s-1}. Hence

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =D1|t|s1𝑑x=qd/2|t|s.absentsubscriptsubscript𝐷1superscript𝑡𝑠1differential-d𝑥superscript𝑞𝑑2superscript𝑡𝑠\displaystyle=\textstyle{\int}_{D_{1}}|t|^{-s-1}dx=q^{-d/2}|t|^{-s}.

Since D2={xFv||x|>|t|}subscript𝐷2conditional-set𝑥subscript𝐹𝑣𝑥𝑡D_{2}=\{x\in F_{v}||x|>|t|\}, from (10.19) we have

B2(t)subscript𝐵2𝑡\displaystyle B_{2}(t) =|t|s+1|x|>|t||x|2s2𝑑x=qd/2(1q1)q2s11q2s1|t|s.absentsuperscript𝑡𝑠1subscript𝑥𝑡superscript𝑥2𝑠2differential-d𝑥superscript𝑞𝑑21superscript𝑞1superscript𝑞2𝑠11superscript𝑞2𝑠1superscript𝑡𝑠\displaystyle=|t|^{s+1}\textstyle{\int}_{|x|>|t|}|x|^{-2s-2}{{d}}x=q^{-d/2}(1-q^{-1})\frac{q^{-2s-1}}{1-q^{-2s-1}}|t|^{-s}.

Therefore, we obtain the desired formula by B(t)=B1(t)+B2(t)𝐵𝑡subscript𝐵1𝑡subscript𝐵2𝑡B(t)=B_{1}(t)+B_{2}(t). ∎

Lemma 10.7.

Suppose |τ|=q1𝜏superscript𝑞1|\tau|=q^{-1}, |a|>1𝑎1|a|>1 and set b=a1𝑏superscript𝑎1b=a^{-1}. Let Re(s)>1/2Re𝑠12\operatorname{Re}(s)>-1/2.

  • (i)

    When |t||τb|2𝑡superscript𝜏𝑏2|t|\leqslant|\tau b|^{2},

    B(t)𝐵𝑡\displaystyle B(t) =qd/2|t|s+1|τb|2s1(1+qs1+(1q2s2)(1q2s1)1).absentsuperscript𝑞𝑑2superscript𝑡𝑠1superscript𝜏𝑏2𝑠11superscript𝑞𝑠11superscript𝑞2𝑠2superscript1superscript𝑞2𝑠11\displaystyle=q^{-d/2}|t|^{s+1}|\tau b|^{-2s-1}\left(-1+q^{-s-1}+{(1-q^{-2s-2})}{(1-q^{-2s-1})^{-1}}\right).
  • (ii)

    When |τb|2<|t|1superscript𝜏𝑏2𝑡1|\tau b|^{2}<|t|\leqslant 1,

    B(t)𝐵𝑡\displaystyle B(t) =qd/2([|t|1/2]+|t|s+1[|t|1/2]2s1(1q1)q2s1(1q2s1)1).absentsuperscript𝑞𝑑2delimited-[]superscript𝑡12superscript𝑡𝑠1superscriptdelimited-[]superscript𝑡122𝑠11superscript𝑞1superscript𝑞2𝑠1superscript1superscript𝑞2𝑠11\displaystyle=q^{-d/2}\biggl{(}[|t|^{1/2}]+|t|^{s+1}[|t|^{1/2}]^{-2s-1}{(1-q^{-1})q^{-2s-1}}{(1-q^{-2s-1})^{-1}}\biggr{)}.
  • (iii)

    When 1<|t|1𝑡1<|t|,

    B(t)𝐵𝑡\displaystyle B(t) =qd/2|t|s(1q2s2)(1q2s1)1.absentsuperscript𝑞𝑑2superscript𝑡𝑠1superscript𝑞2𝑠2superscript1superscript𝑞2𝑠11\displaystyle=q^{-d/2}|t|^{-s}{(1-q^{-2s-2})}{(1-q^{-2s-1})^{-1}}.
Proof.

Note that |τb|<|τ|=q1<1𝜏𝑏𝜏superscript𝑞11|\tau b|<|\tau|=q^{-1}<1. As in the proof of Lemma 10.6, we write B(t)=B1(t)+B2(t)𝐵𝑡subscript𝐵1𝑡subscript𝐵2𝑡B(t)=B_{1}(t)+B_{2}(t), Bi(t)=Dif(x)𝑑xsubscript𝐵𝑖𝑡subscriptsubscript𝐷𝑖𝑓𝑥differential-d𝑥B_{i}(t)=\int_{D_{i}}f(x)\,{{d}}x with D1={xFv||x|max(|τb|,|t|)}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝜏𝑏𝑡D_{1}=\{x\in F_{v}|\,|x|\leqslant\max(|\tau b|,|t|)\,\} and D2=FvD1subscript𝐷2subscript𝐹𝑣subscript𝐷1D_{2}=F_{v}-D_{1}. For xD2𝑥subscript𝐷2x\in D_{2}, we have |τb|<|x|𝜏𝑏𝑥|\tau b|<|x| and |x|>|t|𝑥𝑡|x|>|t|, by which |t1(τb2x2)|=|t|1|x|2>|t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|=|t|^{-1}|x|^{2}>|t|. Thus f(x)=max(|1x|,|1+x|,|t|1|x|2)s1f(x)=\max(|1-x|,|1+x|,|t|^{-1}|x|^{2})^{-s-1} for xD2𝑥subscript𝐷2x\in D_{2}. We have f(x)=max(|1+x|,|1x|,|t|,|t1(τb2x2)|)s1f(x)=\max(|1+x|,|1-x|,|t|,|t^{-1}(\tau b^{2}-x^{2})|)^{-s-1} for xD1𝑥subscript𝐷1x\in D_{1}.

(i) Suppose |t||τb|𝑡𝜏𝑏|t|\leqslant|\tau b|. For xD1𝔭𝑥subscript𝐷1𝔭x\in D_{1}\cap{\mathfrak{p}}, we have |1±x|=1plus-or-minus1𝑥1|1\pm x|=1 and |t||t|1|τ2b2|<|t|1|τb2|=|t1(τb2x2)|𝑡superscript𝑡1superscript𝜏2superscript𝑏2superscript𝑡1𝜏superscript𝑏2superscript𝑡1𝜏superscript𝑏2superscript𝑥2|t|\leqslant|t|^{-1}|\tau^{2}b^{2}|<|t|^{-1}|\tau b^{2}|=|t^{-1}(\tau b^{2}-x^{2})|; thus f(x)=max(1,|t|1|τb2|)s1f(x)=\max(1,|t|^{-1}|\tau b^{2}|)^{-s-1}. The set D1(Fv𝔭)subscript𝐷1subscript𝐹𝑣𝔭D_{1}\cap(F_{v}-{\mathfrak{p}}) is empty due to |τb|<1𝜏𝑏1|\tau b|<1. Hence

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =|x||τb||x|<1max(1,|t|1|τb2|)s1dx=qd/2|τb|max(1,|t|1|τb2|)s1.\displaystyle=\textstyle{\int}_{\begin{subarray}{c}|x|\leqslant|\tau b|\\ |x|<1\end{subarray}}\max(1,|t|^{-1}|\tau b^{2}|)^{-s-1}{{d}}x=q^{-d/2}|\tau b|\max(1,|t|^{-1}|\tau b^{2}|)^{-s-1}.

For xD2𝔭𝑥subscript𝐷2𝔭x\in D_{2}\cap{\mathfrak{p}}, we have |1±x|=1plus-or-minus1𝑥1|1\pm x|=1 and f(x)=max(1,|t|1|x|2)s1f(x)=\max(1,|t|^{-1}|x|^{2})^{-s-1}. For xD2𝔬×𝑥subscript𝐷2superscript𝔬x\in D_{2}\cap\mathfrak{o}^{\times}, we have |1±x|1<|τb|1|t|1=|t|1|x|2plus-or-minus1𝑥1superscript𝜏𝑏1superscript𝑡1superscript𝑡1superscript𝑥2|1\pm x|\leqslant 1<|\tau b|^{-1}\leqslant|t|^{-1}=|t|^{-1}|x|^{2}; thus f(x)=|t|s+1𝑓𝑥superscript𝑡𝑠1f(x)=|t|^{s+1}. For xD2(Fv𝔬)𝑥subscript𝐷2subscript𝐹𝑣𝔬x\in D_{2}\cap(F_{v}-\mathfrak{o}), we have |t||τb|<1<|x|𝑡𝜏𝑏1𝑥|t|\leqslant|\tau b|<1<|x|, which implies |x|<|t|1|x|2𝑥superscript𝑡1superscript𝑥2|x|<|t|^{-1}|x|^{2}; thus f(x)=(|t|1|x|2)s1𝑓𝑥superscriptsuperscript𝑡1superscript𝑥2𝑠1f(x)=(|t|^{-1}|x|^{2})^{-s-1}. Therefore,

B2(t)=subscript𝐵2𝑡absent\displaystyle B_{2}(t)= {xD2𝔭|t|1/2<|x|(|t|1|x|2)s1𝑑x+xD2𝔭|t|1/2|x|𝑑x+D2𝔬×|t|s+1𝑑x+D2(Fv𝔬)(|t|1|x|2)s1𝑑x}subscript𝑥subscript𝐷2𝔭superscript𝑡12𝑥superscriptsuperscript𝑡1superscript𝑥2𝑠1differential-d𝑥subscript𝑥subscript𝐷2𝔭superscript𝑡12𝑥differential-d𝑥subscriptsubscript𝐷2superscript𝔬superscript𝑡𝑠1differential-d𝑥subscriptsubscript𝐷2subscript𝐹𝑣𝔬superscriptsuperscript𝑡1superscript𝑥2𝑠1differential-d𝑥\displaystyle\biggl{\{}\textstyle{\int}_{\begin{subarray}{c}x\in D_{2}\cap{\mathfrak{p}}\\ |t|^{1/2}<|x|\end{subarray}}(|t|^{-1}|x|^{2})^{-s-1}{{d}}x+\textstyle{\int}_{\begin{subarray}{c}x\in D_{2}\cap{\mathfrak{p}}\\ |t|^{1/2}\geqslant|x|\end{subarray}}{{d}}x+\textstyle{\int}_{D_{2}\cap\mathfrak{o}^{\times}}|t|^{s+1}{{d}}x+\textstyle{\int}_{D_{2}\cap(F_{v}-\mathfrak{o})}(|t|^{-1}|x|^{2})^{-s-1}{{d}}x\biggr{\}}
=\displaystyle= |τb|z+1{|t|s+1max(|τb|,[|t|1/2])<|x|<1|x|2s2dx+|τb|<|x|[|t|1/2]dx\displaystyle|\tau b|^{z+1}\biggl{\{}|t|^{s+1}\textstyle{\int}_{\max(|\tau b|,[|t|^{1/2}])<|x|<1}|x|^{-2s-2}{{d}}x+\int_{|\tau b|<|x|\leqslant[|t|^{1/2}]}{{d}}x
+|t|s+1|x|=1dx+|t|s+11<|x||x|2s2dx}\displaystyle+|t|^{s+1}\textstyle{\int}_{|x|=1}{{d}}x+|t|^{s+1}\textstyle{\int}_{1<|x|}|{x}|^{-2s-2}{{d}}x\biggr{\}}
=\displaystyle= qd/2(1q1){q2s1max(|τb|,[|t|1/2])2s1q2s11q2s1×|t|s+1\displaystyle q^{-d/2}(1-q^{-1})\biggl{\{}\frac{q^{-2s-1}\max(|\tau b|,[|t|^{1/2}])^{-2s-1}-q^{-2s-1}}{1-q^{-2s-1}}\times|t|^{s+1}
+δ(|τb|<|t|1/2)[|t|1/2]|τb|1q1+q2s11q2s1×|t|s+1}.\displaystyle+\delta(|\tau b|<|t|^{1/2})\frac{[|t|^{1/2}]-|\tau b|}{1-q^{-1}}+\frac{q^{-2s-1}}{1-q^{-2s-1}}\times|t|^{s+1}\biggr{\}}.

From this,

B2(t)=subscript𝐵2𝑡absent\displaystyle B_{2}(t)= qd/2(1q1){|τb|2s1|t|s+1q2s11q2s1if |τb|2|t|,|t|s+1[|t|1/2]2s1q2s11q2s1+[|t|1/2]|τb|1q1if |τb|2<|t||τb|.superscript𝑞𝑑21superscript𝑞1casessuperscript𝜏𝑏2𝑠1superscript𝑡𝑠1superscript𝑞2𝑠11superscript𝑞2𝑠1if |τb|2|t|otherwisesuperscript𝑡𝑠1superscriptdelimited-[]superscript𝑡122𝑠1superscript𝑞2𝑠11superscript𝑞2𝑠1delimited-[]superscript𝑡12𝜏𝑏1superscript𝑞1if |τb|2<|t||τb|otherwise\displaystyle q^{-d/2}(1-q^{-1})\begin{cases}|\tau b|^{-2s-1}|t|^{s+1}\frac{q^{-2s-1}}{1-q^{-2s-1}}\quad\text{if $|\tau b|^{2}\geqslant|t|$},\\ |t|^{s+1}[|t|^{1/2}]^{-2s-1}\frac{q^{-2s-1}}{1-q^{-2s-1}}+\frac{[|t|^{1/2}]-|\tau b|}{1-q^{-1}}\quad\text{if $|\tau b|^{2}<|t|\leqslant|\tau b|$}.\end{cases}

(ii) Suppose |t|>|τb|𝑡𝜏𝑏|t|>|\tau b|. Then D1={xFv||x||t|}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝑡D_{1}=\{x\in F_{v}|\,|x|\leqslant|t|\}. For xD1𝔭𝑥subscript𝐷1𝔭x\in D_{1}\cap{\mathfrak{p}}, we have |1±x|=1plus-or-minus1𝑥1|1\pm x|=1 and |t1(τb2x2)||t|1max(|τb2|,|x|2)|t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2superscript𝑡1𝜏superscript𝑏2superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|\leqslant|t|^{-1}\max(|\tau b^{2}|,|x|^{2})\leqslant|t|; hence f(x)=max(1,|t|)s1f(x)=\max(1,|t|)^{-s-1}. For xD1(Fv𝔭)𝑥subscript𝐷1subscript𝐹𝑣𝔭x\in D_{1}\cap(F_{v}-{\mathfrak{p}}), |1±x||x||t|plus-or-minus1𝑥𝑥𝑡|1\pm x|\leqslant|x|\leqslant|t| and |t1(τb2x2)||t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|\leqslant|t|; thus f(x)=|t|s1𝑓𝑥superscript𝑡𝑠1f(x)=|t|^{-s-1}. Hence

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =|x|min(q1,|t|)max(1,|t|)s1dx+1|x||t||t|s1dx\displaystyle=\textstyle{\int}_{|x|\leqslant\min(q^{-1},|t|)}\max(1,|t|)^{-s-1}\,{{d}}x+\textstyle{\int}_{1\leqslant|x|\leqslant|t|}|t|^{-s-1}{{d}}x
=qd/2{max(1,|t|)s1min(q1,|t|)+|t|sδ(|t|1)(|t|1)}\displaystyle=q^{-d/2}\{\max(1,|t|)^{-s-1}\,\min(q^{-1},|t|)+|t|^{-s}\delta(|t|\geqslant 1)(|t|-1)\}
=qd/2{|t|(|τb|<|t|1),|t|s(1<|t|).absentsuperscript𝑞𝑑2cases𝑡𝜏𝑏𝑡1otherwisesuperscript𝑡𝑠1𝑡otherwise\displaystyle=q^{-d/2}\begin{cases}|t|\quad(|\tau b|<|t|\leqslant 1),\\ |t|^{-s}\quad(1<|t|).\end{cases}

We have D2={xFv||x|>|t|}subscript𝐷2conditional-set𝑥subscript𝐹𝑣𝑥𝑡D_{2}=\{x\in F_{v}||x|>|t|\}. For xD2𝑥subscript𝐷2x\in D_{2}, we have |τb|<|t|<|x|𝜏𝑏𝑡𝑥|\tau b|<|t|<|x| which in turn yields |τb2|<|x|2𝜏superscript𝑏2superscript𝑥2|\tau b^{2}|<|x|^{2} and |t|<|t|1|x|2𝑡superscript𝑡1superscript𝑥2|t|<|t|^{-1}|x|^{2}. Hence f(x)=max(|1+x|,|1x|,|t|1|x|2)|s1f(x)=\max(|1+x|,|1-x|,|t|^{-1}|x|^{2})|^{-s-1} for xD2𝑥subscript𝐷2x\in D_{2}. We have

B2(t)subscript𝐵2𝑡\displaystyle B_{2}(t) =|t|<|x|<1max(1,|t|1|x|2)s1dx+|x|=1δ(|t|<1)|t|s+1dx\displaystyle=\textstyle{\int}_{|t|<|x|<1}\max(1,|t|^{-1}|x|^{2})^{-s-1}{{d}}x+\textstyle{\int}_{|x|=1}\delta(|t|<1)|t|^{s+1}{{d}}x
+max(1,|t|)<|x|(|t|1|x|2)s1𝑑xsubscript1𝑡𝑥superscriptsuperscript𝑡1superscript𝑥2𝑠1differential-d𝑥\displaystyle+\textstyle{\int}_{\max(1,|t|)<|x|}(|t|^{-1}|x|^{2})^{-s-1}{{d}}x
=qd/2(1q1)|τb|z+1{δ(|t|<1)|t|s+1q2s1([|t|1/2]2s1q2s+1)1q2s1\displaystyle=q^{-d/2}(1-q^{-1})|\tau b|^{z+1}\biggl{\{}\delta(|t|<1)|t|^{s+1}\frac{q^{-2s-1}([|t|^{1/2}]^{-2s-1}-q^{2s+1})}{1-q^{-2s-1}}
+δ(|t|1)[|t|1/2]|t|1q1+δ(|t|<1)|t|s+1+|t|s+1q2s1max(1,|t|)2s11q2s1}\displaystyle\quad+\delta(|t|\leqslant 1)\frac{[|t|^{1/2}]-|t|}{1-q^{-1}}+\delta(|t|<1)|t|^{s+1}+|t|^{s+1}\frac{q^{-2s-1}\max(1,|t|)^{-2s-1}}{1-q^{-2s-1}}\biggr{\}}
=qd/2(1q1){|t|sq2s11q2s1(1<|t|),|t|s+1[|t|1/2]2s1q2s11q2s1+[|t|1/2]|t|1q1(|τb|<|t|1).absentsuperscript𝑞𝑑21superscript𝑞1casessuperscript𝑡𝑠superscript𝑞2𝑠11superscript𝑞2𝑠11𝑡otherwisesuperscript𝑡𝑠1superscriptdelimited-[]superscript𝑡122𝑠1superscript𝑞2𝑠11superscript𝑞2𝑠1delimited-[]superscript𝑡12𝑡1superscript𝑞1𝜏𝑏𝑡1otherwise\displaystyle=q^{-d/2}(1-q^{-1})\begin{cases}|t|^{-s}\frac{q^{-2s-1}}{1-q^{-2s-1}}\quad(1<|t|),\\ |t|^{s+1}[|t|^{1/2}]^{-2s-1}\frac{q^{-2s-1}}{1-q^{-2s-1}}+\frac{[|t|^{1/2}]-|t|}{1-q^{-1}}\quad(|\tau b|<|t|\leqslant 1).\end{cases}

Here to have the second equality, we split the integral over |t|<|x|<1𝑡𝑥1|t|<|x|<1 to those over |t|1/2<|x|<1superscript𝑡12𝑥1|t|^{1/2}<|x|<1 and over |t|<|x||t|1/2𝑡𝑥superscript𝑡12|t|<|x|\leqslant|t|^{1/2}.

From the evaluations of B1(t)subscript𝐵1𝑡B_{1}(t) and B2(t)subscript𝐵2𝑡B_{2}(t), by B(t)=B1(t)+B2(t)𝐵𝑡subscript𝐵1𝑡subscript𝐵2𝑡B(t)=B_{1}(t)+B_{2}(t) we are done. ∎

The following is given by a direct computation.

Lemma 10.8.

If α,β𝛼𝛽\alpha,\beta\in\mathbb{C}, then

|τb|2<|t|1[|t|1/2]α|t|βd×t=qd/2(1+qvαβ1qα2β+1qαβ1qα2β|τb|α+2β).subscriptsuperscript𝜏𝑏2𝑡1superscriptdelimited-[]superscript𝑡12𝛼superscript𝑡𝛽superscript𝑑𝑡superscript𝑞𝑑21superscriptsubscript𝑞𝑣𝛼𝛽1superscript𝑞𝛼2𝛽1superscript𝑞𝛼𝛽1superscript𝑞𝛼2𝛽superscript𝜏𝑏𝛼2𝛽\int_{|\tau b|^{2}<|t|\leqslant 1}[|t|^{1/2}]^{\alpha}|t|^{\beta}{{d}}^{\times}t=q^{-d/2}\left(\frac{1+q_{v}^{-\alpha-\beta}}{1-q^{-\alpha-2\beta}}+\frac{-1-q^{-\alpha-\beta}}{1-q^{-\alpha-2\beta}}|\tau b|^{\alpha+2\beta}\right).
Lemma 10.9.

Let ordv(τ)=1subscriptord𝑣𝜏1\operatorname{ord}_{v}(\tau)=1. Set b=a1𝑏superscript𝑎1b=a^{-1}. Suppose Re(s)>(|Re(z)|1)/2Re𝑠Re𝑧12\operatorname{Re}(s)>(|\operatorname{Re}(z)|-1)/2. For |a|1𝑎1|a|\leqslant 1,

Fv×B(t)|t|z12d×t=qd|b|s+z12(1+qz12)ζFv(s+1+z12)ζFv(s+1+z12)ζFv(s+1).subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡superscript𝑡𝑧12superscript𝑑𝑡superscript𝑞𝑑superscript𝑏𝑠𝑧121superscript𝑞𝑧12subscript𝜁subscript𝐹𝑣𝑠1𝑧12subscript𝜁subscript𝐹𝑣𝑠1𝑧12subscript𝜁subscript𝐹𝑣𝑠1\int_{F_{v}^{\times}}B(t)|t|^{\frac{z-1}{2}}d^{\times}t=q^{-d}|b|^{-s+\frac{z-1}{2}}(1+q^{\frac{-z-1}{2}})\frac{\zeta_{F_{v}}(s+1+\frac{z-1}{2})\zeta_{F_{v}}(s+1+\frac{-z-1}{2})}{\zeta_{F_{v}}(s+1)}.

For |a|>1𝑎1|a|>1,

Fv×B(t)|t|(z1)/2d×t=qd1+qz+12ζFv(s+1)(ζFv(z)ζFv(s+1+z12)+ζFv(z)ζFv(s+1+z12)|b|z).subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡superscript𝑡𝑧12superscript𝑑𝑡superscript𝑞𝑑1superscript𝑞𝑧12subscript𝜁subscript𝐹𝑣𝑠1subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣𝑠1𝑧12subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣𝑠1𝑧12superscript𝑏𝑧\int_{F_{v}^{\times}}B(t)|t|^{(z-1)/2}d^{\times}t=q^{-d}\frac{1+q^{-\frac{z+1}{2}}}{\zeta_{F_{v}}(s+1)}\left(\zeta_{F_{v}}(z)\zeta_{F_{v}}\left(s+1+\tfrac{-z-1}{2}\right)+\zeta_{F_{v}}(-z)\zeta_{F_{v}}\left(s+1+\tfrac{z-1}{2}\right)|b|^{z}\right).
Proof.

This is proved by a straightforward computation by means of Lemmas 10.6, 10.7 and 10.8. ∎

We obtain the formula for a=t2mv0𝑎𝑡2subscript𝑚𝑣0a=\frac{t}{2m_{v}}\neq 0 in Theorem 7.9 (5) for the case τ𝔭𝔭2𝜏𝔭superscript𝔭2\tau\in{\mathfrak{p}}-{\mathfrak{p}}^{2} from (10.18) applying Lemma 10.9 combined with Lemma 7.4, Lemma 7.11 (2) and Proposition 7.12 (2). We note that L(s,εΔ,v)=1𝐿𝑠subscript𝜀Δ𝑣1L(s,\varepsilon_{\Delta,v})=1.

Next we consider the case when τ=Δv0𝜏superscriptsubscriptΔ𝑣0\tau=\Delta_{v}^{0} is a non-square unit.

Lemma 10.10.

Suppose τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in\mathfrak{o}^{\times}-(\mathfrak{o}^{\times})^{2} and |a|>1𝑎1|a|>1. Set b=a1𝑏superscript𝑎1b=a^{-1}. Suppose Re(s)>1/2Re𝑠12\operatorname{Re}(s)>-1/2.

  • (i)

    When |t||b|2𝑡superscript𝑏2|t|\leqslant|b|^{2},

    B(t)𝐵𝑡\displaystyle B(t) =qd/2|t|s+1|b|2s1(1q2s2)(1q2s1)1.absentsuperscript𝑞𝑑2superscript𝑡𝑠1superscript𝑏2𝑠11superscript𝑞2𝑠2superscript1superscript𝑞2𝑠11\displaystyle=q^{-d/2}|t|^{s+1}|b|^{-2s-1}{(1-q^{-2s-2})}{(1-q^{-2s-1})^{-1}}.
  • (ii)

    When |b|2<|t|1superscript𝑏2𝑡1|b|^{2}<|t|\leqslant 1,

    B(t)𝐵𝑡\displaystyle B(t) =qd/2{(1q1)|t|s+1[|t|1/2]2s1q2s1(1q2s1)1+[|t|1/2]}.absentsuperscript𝑞𝑑21superscript𝑞1superscript𝑡𝑠1superscriptdelimited-[]superscript𝑡122𝑠1superscript𝑞2𝑠1superscript1superscript𝑞2𝑠11delimited-[]superscript𝑡12\displaystyle=q^{-d/2}\left\{(1-q^{-1})|t|^{s+1}[|t|^{1/2}]^{-2s-1}{q^{-2s-1}}{(1-q^{-2s-1})^{-1}}+[|t|^{1/2}]\right\}.
  • (iii)

    When 1<|t|1𝑡1<|t|,

    B(t)𝐵𝑡\displaystyle B(t) =qd/2|t|s(1q2s2)(1q2s1)1.absentsuperscript𝑞𝑑2superscript𝑡𝑠1superscript𝑞2𝑠2superscript1superscript𝑞2𝑠11\displaystyle=q^{-d/2}|t|^{-s}{(1-q^{-2s-2})}{(1-q^{-2s-1})^{-1}}.
Proof.

Let D1={xFv||x|max(|b|,|t|)}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝑏𝑡D_{1}=\{x\in F_{v}|\,|x|\leqslant\max(|b|,|t|)\} and D2={xFv||x|>max(|b|,|t|)}subscript𝐷2conditional-set𝑥subscript𝐹𝑣𝑥𝑏𝑡D_{2}=\{x\in F_{v}|\,|x|>\max(|b|,|t|)\}, and set Bi(t)=Dif(x)𝑑xsubscript𝐵𝑖𝑡subscriptsubscript𝐷𝑖𝑓𝑥differential-d𝑥B_{i}(t)=\int_{D_{i}}f(x){{d}}x for i=1,2𝑖12i=1,2 to write B(t)𝐵𝑡B(t) as the sum B1(t)+B2(t)subscript𝐵1𝑡subscript𝐵2𝑡B_{1}(t)+B_{2}(t), where

f(x)=max(|1+x|,|1x|,|t|,|t1(τb2x2)|)s1,xFv.f(x)=\max(|1+x|,|1-x|,|t|,|t^{-1}(\tau b^{2}-x^{2})|)^{-s-1},x\in F_{v}.

(i) Suppose |t||b|𝑡𝑏|t|\leqslant|b|. Then D1={xFv||x||b|}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝑏D_{1}=\{x\in F_{v}||x|\leqslant|b|\}. Since |b|<1𝑏1|b|<1, the set D1(F𝔭)subscript𝐷1𝐹𝔭D_{1}\cap(F-{\mathfrak{p}}) is empty. If xD1𝔭𝑥subscript𝐷1𝔭x\in D_{1}\cap{\mathfrak{p}}, then |1±x|=1plus-or-minus1𝑥1|1\pm x|=1 and |t||t|1|b|2=|t1(τb2x2)|𝑡superscript𝑡1superscript𝑏2superscript𝑡1𝜏superscript𝑏2superscript𝑥2|t|\leqslant|t|^{-1}|b|^{2}=|t^{-1}(\tau b^{2}-x^{2})| by Lemma 7.1; hence f(x)=max(1,|t|1|b|2)s1f(x)=\max(1,|t|^{-1}|b|^{2})^{-s-1}. Thus

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =max(1,|t|1|b|2)s1|x||b|dx=qd/2|b|max(1,|t|1|b|2)s1.\displaystyle=\max(1,|t|^{-1}|b|^{2})^{-s-1}\textstyle{\int}_{|x|\leqslant|b|}{{d}}x=q^{-d/2}|b|\max(1,|t|^{-1}|b|^{2})^{-s-1}.

For xD2𝑥subscript𝐷2x\in D_{2}, we have |x|>|b||t|𝑥𝑏𝑡|x|>|b|\geqslant|t|; hence |t1(τb2x2)|=|t|1|x|2>|t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|=|t|^{-1}|x|^{2}>|t| and f(x)=max(|1x|,|1+x|,|t|1|x|2)s1f(x)=\max(|1-x|,|1+x|,|t|^{-1}|x|^{2})^{-s-1}. Thus B2(t)subscript𝐵2𝑡B_{2}(t) equals

|b|<|x|<1max(1,|t|1|x|2)s1dx+|x|=1|t|s+1dx+|x|>1max(|x|,|t|1|x|2)s1dx\displaystyle\textstyle{\int}_{|b|<|x|<1}\max(1,|t|^{-1}|x|^{2})^{-s-1}{{d}}x+\textstyle{\int}_{|x|=1}|t|^{s+1}{{d}}x+\textstyle{\int}_{|x|>1}\max(|x|,|t|^{-1}|x|^{2})^{-s-1}{{d}}x
=\displaystyle= |t|s+1max(|b|,|t|1/2)<|x|<1|x|2s2𝑑x+|b|<|x||t|1/2𝑑xsuperscript𝑡𝑠1subscript𝑏superscript𝑡12𝑥1superscript𝑥2𝑠2differential-d𝑥subscript𝑏𝑥superscript𝑡12differential-d𝑥\displaystyle|t|^{s+1}\textstyle{\int}_{\max(|b|,|t|^{1/2})<|x|<1}|x|^{-2s-2}{{d}}x+\textstyle{\int}_{|b|<|x|\leqslant|t|^{1/2}}{{d}}x
+|t|s+1qd/2(1q1)+|t|s+1|x|>1|x|2s2𝑑xsuperscript𝑡𝑠1superscript𝑞𝑑21superscript𝑞1superscript𝑡𝑠1subscript𝑥1superscript𝑥2𝑠2differential-d𝑥\displaystyle+|t|^{s+1}\,q^{-d/2}(1-q^{-1})+|t|^{s+1}\textstyle{\int}_{|x|>1}|x|^{-2s-2}{{d}}x
=\displaystyle= |t|s+1max(|b|,[|t|1/2)<|x|q1|x|2s2𝑑x+|b|<|x|[|t|1/2]𝑑x\displaystyle|t|^{s+1}\textstyle{\int}_{\max(|b|,[|t|^{1/2})<|x|\leqslant q^{-1}}|x|^{-2s-2}{{d}}x+\textstyle{\int}_{|b|<|x|\leqslant[|t|^{1/2}]}{{d}}x
+|t|s+1×qd/2(1q1)+|t|s+1×qd/2(1q1)q2s11q2s1superscript𝑡𝑠1superscript𝑞𝑑21superscript𝑞1superscript𝑡𝑠1superscript𝑞𝑑21superscript𝑞1superscript𝑞2𝑠11superscript𝑞2𝑠1\displaystyle+|t|^{s+1}\times q^{-d/2}(1-q^{-1})+|t|^{s+1}\times q^{-d/2}(1-q^{-1})\tfrac{q^{-2s-1}}{1-q^{-2s-1}}
=\displaystyle= qd/2(1q1){|t|s+1q2s1(max(|b|,[|t|1/2])2s1q2s+1)1q2s1+δ(|b|2|t|)[|t|1/2]|b|1q1+|t|s+111q2s1}.\displaystyle\,q^{-d/2}(1-q^{-1})\biggl{\{}|t|^{s+1}\tfrac{q^{-2s-1}(\max(|b|,[|t|^{1/2}])^{-2s-1}-q^{2s+1})}{1-q^{-2s-1}}+\delta(|b|^{2}\leqslant|t|)\tfrac{[|t|^{1/2}]-|b|}{1-q^{-1}}+|t|^{s+1}\tfrac{1}{1-q^{-2s-1}}\biggr{\}}.

(ii) Suppose |t|>|b|𝑡𝑏|t|>|b|. Then D1={xFv||x||t|}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝑡D_{1}=\{x\in F_{v}|\,|x|\leqslant|t|\}. If xD1𝔭𝑥subscript𝐷1𝔭x\in D_{1}\cap{\mathfrak{p}}, then |1±x|=1plus-or-minus1𝑥1|1\pm x|=1, |t1(τb2x2)||t|1max(|b|2,|x|2)|t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑏2superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|\leqslant|t|^{-1}\max(|b|^{2},|x|^{2})\leqslant|t|; hence f(x)=max(1,|t|)s1f(x)=\max(1,|t|)^{-s-1}. If xD1(Fv𝔭)𝑥subscript𝐷1subscript𝐹𝑣𝔭x\in D_{1}\cap(F_{v}-{\mathfrak{p}}), then |1±x||x||t|plus-or-minus1𝑥𝑥𝑡|1\pm x|\leqslant|x|\leqslant|t| and |t1(τb2x2)||t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|\leqslant|t| as above; hence f(x)=|t|s1𝑓𝑥superscript𝑡𝑠1f(x)=|t|^{-s-1}. Thus

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =max(1,|t|)s1xD1𝔭dx+|t|s11|x||t|dx=qd/2{|t|(|t|<1),|t|s(|t|1).\displaystyle=\max(1,|t|)^{-s-1}\textstyle{\int}_{x\in D_{1}\cap{\mathfrak{p}}}{{d}}x+|t|^{-s-1}\textstyle{\int}_{1\leqslant|x|\leqslant|t|}{{d}}x=q^{-d/2}\begin{cases}|t|\quad(|t|<1),\\ |t|^{-s}\quad(|t|\geqslant 1).\end{cases}

For all xD2𝑥subscript𝐷2x\in D_{2}, we have |t|<|t|1|x|2=|t1(τb2x2)|𝑡superscript𝑡1superscript𝑥2superscript𝑡1𝜏superscript𝑏2superscript𝑥2|t|<|t|^{-1}|x|^{2}=|t^{-1}(\tau b^{2}-x^{2})|. If xD2𝔭𝑥subscript𝐷2𝔭x\in D_{2}\cap{\mathfrak{p}}, we have |1±x|=1plus-or-minus1𝑥1|1\pm x|=1; thus f(x)=max(1,|t|1|x|2)s1f(x)=\max(1,|t|^{-1}|x|^{2})^{-s-1}. If xD2𝔬×𝑥subscript𝐷2superscript𝔬x\in D_{2}\cap\mathfrak{o}^{\times}, we have |1±x|1<|t|1=|t|1|x|2plus-or-minus1𝑥1superscript𝑡1superscript𝑡1superscript𝑥2|1\pm x|\leqslant 1<|t|^{-1}=|t|^{-1}|x|^{2}; hence f(x)=|t|s+1𝑓𝑥superscript𝑡𝑠1f(x)=|t|^{s+1}. Note that D2𝔬×subscript𝐷2superscript𝔬D_{2}\cap\mathfrak{o}^{\times} is empty unless |t|<1𝑡1|t|<1. If xD2(Fv𝔬)𝑥subscript𝐷2subscript𝐹𝑣𝔬x\in D_{2}\cap(F_{v}-\mathfrak{o}), then |t|<|x|𝑡𝑥|t|<|x| and |1±x|=|x|<|t|1|x|2plus-or-minus1𝑥𝑥superscript𝑡1superscript𝑥2|1\pm x|=|x|<|t|^{-1}|x|^{2}; hence f(x)=(|t|1|x|2)s1𝑓𝑥superscriptsuperscript𝑡1superscript𝑥2𝑠1f(x)=(|t|^{-1}|x|^{2})^{-s-1}. Thus,

B2(t)=subscript𝐵2𝑡absent\displaystyle B_{2}(t)= δ(|t|<1)|t|<|x|<1max(1,|t|1|x|2)s1dx+δ(|t|<1)|t|s+1|x|=1dx\displaystyle\delta(|t|<1)\textstyle{\int}_{|t|<|x|<1}\max(1,|t|^{-1}|x|^{2})^{-s-1}{{d}}x+\delta(|t|<1)|t|^{s+1}\textstyle{\int}_{|x|=1}{{d}}x
+|t|s+1|x|>max(1,|t|)|x|2s2𝑑xsuperscript𝑡𝑠1subscript𝑥1𝑡superscript𝑥2𝑠2differential-d𝑥\displaystyle+|t|^{s+1}\textstyle{\int}_{|x|>\max(1,|t|)}|x|^{-2s-2}{{d}}x
=\displaystyle= δ(|t|<1)|t|s+1|t|1/2<|x|<1|x|2s2𝑑x+δ(|t|<1)|t|<|x||t|1/2𝑑x𝛿𝑡1superscript𝑡𝑠1subscriptsuperscript𝑡12𝑥1superscript𝑥2𝑠2differential-d𝑥𝛿𝑡1subscript𝑡𝑥superscript𝑡12differential-d𝑥\displaystyle\delta(|t|<1)|t|^{s+1}\textstyle{\int}_{|t|^{1/2}<|x|<1}|x|^{-2s-2}{{d}}x+\delta(|t|<1)\textstyle{\int}_{|t|<|x|\leqslant|t|^{1/2}}{{d}}x
+δ(|t|<1)|t|s+1qd/2(1q1)+|t|s+1qd/2(1q1)q2s1max(1,|t|)2s11q2s1\displaystyle\quad+\delta(|t|<1)|t|^{s+1}\,q^{-d/2}(1-q^{-1})+|t|^{s+1}\,q^{-d/2}(1-q^{-1})\tfrac{q^{-2s-1}\max(1,|t|)^{-2s-1}}{1-q^{-2s-1}}
=\displaystyle= qd/2(1q1){δ(|t|<1)|t|s+1q2s1[|t|1/2]2s111q2s1+δ(|t|<1)[|t|1/2]|t|1q1\displaystyle q^{-d/2}(1-q^{-1})\biggl{\{}\delta(|t|<1)|t|^{s+1}\tfrac{q^{-2s-1}[|t|^{1/2}]^{-2s-1}-1}{1-q^{-2s-1}}+\delta(|t|<1)\tfrac{[|t|^{1/2}]-|t|}{1-q^{-1}}
+δ(|t|<1)|t|s+1+|t|s+1q2s1max(1,|t|)2s11q2s1}\displaystyle+\delta(|t|<1)|t|^{s+1}+|t|^{s+1}\tfrac{q^{-2s-1}\max(1,|t|)^{-2s-1}}{1-q^{-2s-1}}\biggr{\}}
=\displaystyle= qd/2(1q1){|t|sq2s11q2s1(|t|>1),|t|s+1[|t|1/2]2s1q2s11q2s1+[|t|1/2]|t|1q1(|b|<|t|1).superscript𝑞𝑑21superscript𝑞1casessuperscript𝑡𝑠superscript𝑞2𝑠11superscript𝑞2𝑠1𝑡1otherwisesuperscript𝑡𝑠1superscriptdelimited-[]superscript𝑡122𝑠1superscript𝑞2𝑠11superscript𝑞2𝑠1delimited-[]superscript𝑡12𝑡1superscript𝑞1𝑏𝑡1otherwise\displaystyle q^{-d/2}(1-q^{-1})\begin{cases}|t|^{-s}\tfrac{q^{-2s-1}}{1-q^{-2s-1}}\quad(|t|>1),\\ |t|^{s+1}[|t|^{1/2}]^{-2s-1}\tfrac{q^{-2s-1}}{1-q^{-2s-1}}+\tfrac{[|t|^{1/2}]-|t|}{1-q^{-1}}\quad(|b|<|t|\leqslant 1).\end{cases}

From the evaluations of B1(t)subscript𝐵1𝑡B_{1}(t) and B2(t)subscript𝐵2𝑡B_{2}(t), we are done. ∎

Lemma 10.11.

Suppose τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in\mathfrak{o}^{\times}-(\mathfrak{o}^{\times})^{2} and |a|1𝑎1|a|\leqslant 1. Set b=a1𝑏superscript𝑎1b=a^{-1}. If Re(s)>1/2Re𝑠12\operatorname{Re}(s)>-1/2,

B(t)𝐵𝑡\displaystyle B(t) =qd/2(1q2s2)(1q2s1)1{|t|s+1|b|2s1,(|t||b|),|t|s,(|b|<|t|).absentsuperscript𝑞𝑑21superscript𝑞2𝑠2superscript1superscript𝑞2𝑠11casessuperscript𝑡𝑠1superscript𝑏2𝑠1𝑡𝑏otherwisesuperscript𝑡𝑠𝑏𝑡otherwise\displaystyle=q^{-d/2}{(1-q^{-2s-2})}{(1-q^{-2s-1})^{-1}}\begin{cases}|t|^{s+1}|b|^{-2s-1},\quad(|t|\leqslant|b|),\\ |t|^{-s},\quad(|b|<|t|).\end{cases}
Proof.

Let Disubscript𝐷𝑖D_{i} and Bi(t)subscript𝐵𝑖𝑡B_{i}(t) (i=1,2)𝑖12(i=1,2) be as in the proof of Lemma 10.10. For xD2𝑥subscript𝐷2x\in D_{2}, we have |t|1|x|2>|x|>|b|1superscript𝑡1superscript𝑥2𝑥𝑏1|t|^{-1}|x|^{2}>|x|>|b|\geqslant 1, |x|>|t|𝑥𝑡|x|>|t|, |1±x|=|x|>|t|plus-or-minus1𝑥𝑥𝑡|1\pm x|=|x|>|t| and |t1(τb2x2)|=|t|1|x|2superscript𝑡1𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑥2|t^{-1}(\tau b^{2}-x^{2})|=|t|^{-1}|x|^{2}. Hence f(x)=(|t|1|x|2)s1𝑓𝑥superscriptsuperscript𝑡1superscript𝑥2𝑠1f(x)=(|t|^{-1}|x|^{2})^{-s-1}. Thus

B2(t)subscript𝐵2𝑡\displaystyle B_{2}(t) =|t|s+1max(|b|,|t|)<|x||x|2s2𝑑x=qd/2(1q1)|t|s+1q2s1max(|b|,|t|)2s11q2s1.\displaystyle=|t|^{s+1}\textstyle{\int}_{\max(|b|,|t|)<|x|}|x|^{-2s-2}{{d}}x=q^{-d/2}(1-q^{-1})|t|^{s+1}\frac{q^{-2s-1}\max(|b|,|t|)^{-2s-1}}{1-q^{-2s-1}}.

(i) Suppose |t||b|𝑡𝑏|t|\leqslant|b|. Then D1={xFv||x||b|}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝑏D_{1}=\{x\in F_{v}||x|\leqslant|b|\}. If xD1𝑥subscript𝐷1x\in D_{1}, then |1±x|max(1,|x|)|b||t|1|b|2plus-or-minus1𝑥1𝑥𝑏superscript𝑡1superscript𝑏2|1\pm x|\leqslant\max(1,|x|)\leqslant|b|\leqslant|t|^{-1}|b|^{2} and |t1(τb2x2)|=|t|1|b|2|τb2x2|=|t|1|b|2superscript𝑡1𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑏2𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑏2|t^{-1}(\tau b^{2}-x^{2})|=|t|^{-1}|b|^{2}|\tau-b^{-2}x^{2}|=|t|^{-1}|b|^{2} because τ𝜏\tau is not congruent to a square modulo 𝔭𝔭{\mathfrak{p}} by assumption (Lemma 7.1); hence f(x)=(|t|1|b|2)s1𝑓𝑥superscriptsuperscript𝑡1superscript𝑏2𝑠1f(x)=(|t|^{-1}|b|^{2})^{-s-1}. Thus

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =|x||b|(|t|1|b|2)s1𝑑x=qd/2|t|s+1|b|2s1.absentsubscript𝑥𝑏superscriptsuperscript𝑡1superscript𝑏2𝑠1differential-d𝑥superscript𝑞𝑑2superscript𝑡𝑠1superscript𝑏2𝑠1\displaystyle=\textstyle{\int}_{|x|\leqslant|b|}(|t|^{-1}|b|^{2})^{-s-1}{{d}}x=q^{-d/2}|t|^{s+1}|b|^{-2s-1}.

(ii) Suppose |t|>|b|𝑡𝑏|t|>|b|. Then |t|>1𝑡1|t|>1 and D1={xFv||x||t|}subscript𝐷1conditional-set𝑥subscript𝐹𝑣𝑥𝑡D_{1}=\{x\in F_{v}||x|\leqslant|t|\}. If xD1𝑥subscript𝐷1x\in D_{1}, then we have |1±x|max(1,|x|)|t|plus-or-minus1𝑥1𝑥𝑡|1\pm x|\leqslant\max(1,|x|)\leqslant|t| and |t1(τb2x2)||t|1max(|b|2,|x|2)|t|superscript𝑡1𝜏superscript𝑏2superscript𝑥2superscript𝑡1superscript𝑏2superscript𝑥2𝑡|t^{-1}(\tau b^{2}-x^{2})|\leqslant|t|^{-1}\max(|b|^{2},|x|^{2})\leqslant|t|; hence f(x)=|t|s1𝑓𝑥superscript𝑡𝑠1f(x)=|t|^{-s-1}. Thus,

B1(t)subscript𝐵1𝑡\displaystyle B_{1}(t) =|t|s1|x||t|𝑑x=qd/2|t|s.absentsuperscript𝑡𝑠1subscript𝑥𝑡differential-d𝑥superscript𝑞𝑑2superscript𝑡𝑠\displaystyle=|t|^{-s-1}\textstyle{\int}_{|x|\leqslant|t|}{{d}}x=q^{-d/2}|t|^{-s}.

Having evaluations of Bi(t)subscript𝐵𝑖𝑡B_{i}(t) and B(t)=B1(t)+B2(t)𝐵𝑡subscript𝐵1𝑡subscript𝐵2𝑡B(t)=B_{1}(t)+B_{2}(t), we are done. ∎

Lemma 10.12.

Let τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in{\mathfrak{o}}^{\times}-({\mathfrak{o}}^{\times})^{2}. Suppose Re(s)>(|Re(z)|1)/2Re𝑠Re𝑧12\operatorname{Re}(s)>(|\operatorname{Re}(z)|-1)/2. For |a|1𝑎1|a|\leqslant 1,

Fv×B(t)|t|z12d×t=qd|b|s+z12(1qs1)ζFv(s+1+z12)ζFv(s+1+z12)LFv(s+1,εΔ,v).subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡superscript𝑡𝑧12superscript𝑑𝑡superscript𝑞𝑑superscript𝑏𝑠𝑧121superscript𝑞𝑠1subscript𝜁subscript𝐹𝑣𝑠1𝑧12subscript𝜁subscript𝐹𝑣𝑠1𝑧12subscript𝐿subscript𝐹𝑣𝑠1subscript𝜀Δ𝑣\int_{F_{v}^{\times}}B(t)|t|^{\frac{z-1}{2}}d^{\times}t=q^{-d}|b|^{-s+\frac{z-1}{2}}(1-q^{-s-1})\frac{\zeta_{F_{v}}(s+1+\frac{z-1}{2})\zeta_{F_{v}}(s+1+\frac{-z-1}{2})}{L_{F_{v}}(s+1,\varepsilon_{\Delta,v})}.

For |a|>1𝑎1|a|>1,

Fv×B(t)|t|(z1)/2d×t=qd(1qs1)(ζFv(z)ζFv(s+1+z12)LFv(z+12,εΔ,v)+ζFv(z)ζFv(s+1+z12)LFv(z+12,εΔ,v)|b|z).subscriptsuperscriptsubscript𝐹𝑣𝐵𝑡superscript𝑡𝑧12superscript𝑑𝑡superscript𝑞𝑑1superscript𝑞𝑠1subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣𝑠1𝑧12subscript𝐿subscript𝐹𝑣𝑧12subscript𝜀Δ𝑣subscript𝜁subscript𝐹𝑣𝑧subscript𝜁subscript𝐹𝑣𝑠1𝑧12subscript𝐿subscript𝐹𝑣𝑧12subscript𝜀Δ𝑣superscript𝑏𝑧\int_{F_{v}^{\times}}B(t)|t|^{(z-1)/2}d^{\times}t=q^{-d}(1-q^{-s-1})\left(\frac{\zeta_{F_{v}}(z)\zeta_{F_{v}}(s+1+\frac{-z-1}{2})}{L_{F_{v}}(\frac{z+1}{2},\varepsilon_{\Delta,v})}+\frac{\zeta_{F_{v}}(-z)\zeta_{F_{v}}(s+1+\frac{z-1}{2})}{L_{F_{v}}(\frac{-z+1}{2},\varepsilon_{\Delta,v})}|b|^{z}\right).
Proof.

This is proved by a straightforward calculation by means of Lemma 10.10, 10.11, and 10.8. ∎

We obtain the formulafor a=t2mv0𝑎𝑡2subscript𝑚𝑣0a=\frac{t}{2m_{v}}\neq 0 in Theorem 7.9 (5) for the case τ𝔬×(𝔬×)2𝜏superscript𝔬superscriptsuperscript𝔬2\tau\in\mathfrak{o}^{\times}-(\mathfrak{o}^{\times})^{2} from (10.18) applying Lemma 10.12 combined with Lemma 7.4, Lemma 7.11 (1) and Proposition 7.12 (1).

The case a=0𝑎0a=0 is included in the case |a|1𝑎1|a|\leqslant 1. Indeed, we can evaluate 𝔈v(z)(γ^v)superscriptsubscript𝔈𝑣𝑧subscript^𝛾𝑣{\mathfrak{E}}_{v}^{(z)}(\hat{\gamma}_{v}) when 0<|a|10𝑎10<|a|\leqslant 1 as

φ0,v(12)𝔈v(γ^v)=|a2τ|s+12(qs+12qs+12)Fv×B~(a,t)|t|z12d×t,subscript𝜑0𝑣subscript12subscript𝔈𝑣subscript^𝛾𝑣superscriptsuperscript𝑎2𝜏𝑠12superscript𝑞𝑠12superscript𝑞𝑠12subscriptsuperscriptsubscript𝐹𝑣~𝐵𝑎𝑡superscript𝑡𝑧12superscript𝑑𝑡\varphi_{0,v}(1_{2}){\mathfrak{E}}_{v}(\hat{\gamma}_{v})=|a^{2}-\tau|^{\frac{s+1}{2}}(q^{-\frac{s+1}{2}}-q^{\frac{s+1}{2}})\int_{F_{v}^{\times}}{\tilde{B}}(a,t)|t|^{\frac{z-1}{2}}d^{\times}t,

where we set B~(a,t)=|a|s|τ|1B(a,τa1t)~𝐵𝑎𝑡superscript𝑎𝑠superscript𝜏1𝐵𝑎𝜏superscript𝑎1𝑡{\tilde{B}}(a,t)=|a|^{-s}|\tau|^{-1}B(a,\tau a^{-1}t) and we write B(a,t)𝐵𝑎𝑡B(a,t) for B(t)𝐵𝑡B(t) to make dependence on a𝑎a explicit. The function B~(a,t)~𝐵𝑎𝑡\tilde{B}(a,t) is independent of a𝑎a by Lemma 10.6 (i) and Lemma 10.11 (i). Since B~(a,t)~𝐵𝑎𝑡\tilde{B}(a,t) is continuous at a=0𝑎0a=0 by its integral representation, we obtain the formula of B~(0,t)~𝐵0𝑡\tilde{B}(0,t) by limaaB~(a,t)subscript𝑎𝑎~𝐵𝑎𝑡\lim_{a\rightarrow a}\tilde{B}(a,t). Hence the formula in Theorem 7.9 (5) is valid when a=t2mv=0𝑎𝑡2subscript𝑚𝑣0a=\frac{t}{2m_{v}}=0.

Acknowledgements

The second author was supported by Grant-in-Aid for Scientific research (C) 15K04795.

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Shingo SUGIYAMA
Department of Mathematics, College of Science and Technology, Nihon University, Suruga-Dai, Kanda, Chiyoda, Tokyo 101-8308, Japan
E-mail : s-sugiyama@math.cst.nihon-u.ac.jp

Masao TSUZUKI
Department of Science and Technology, Sophia University, Kioi-cho 7-1 Chiyoda-ku Tokyo, 102-8554, Japan
E-mail : m-tsuduk@sophia.ac.jp